CastleData.txt

Castle 1,Castle 2,Castle 3,Castle 4,Castle 5,Castle 6,Castle 7,Castle 8,Castle 9,Castle 10,Why did you choose your troop deployment?
100,0,0,0,0,0,0,0,0,0,"because, I am number one!"
52,2,2,2,2,2,2,12,12,12,"I need to win at least 4 castles to win the game. Any combination of 7 castles wins the game. I assume that the border cases of trying to win 1-7 or 1 and 8-10 will be popular. If possible, I should like to be able to beat either strategy.

One way to do that would be to play minimally on all numbers except for 1. Then I take the ones they don't want, but I also steal castle 1, which is less sought after.

Of course, I lose to the ""10s all around"" strategy, which I imagine will also be popular.

Notice that the key is not beating a randomly generated opponent, but beating the most opponents, which means I want to be able to beat the most popular strategies. Hmm.

The method I've devised will beat ""10s all around"" and has a shot at beating folks who go all in on another strategy. I expect to get beaten a lot, though, by folks who pick a different set of castles they want to win. Oh well. I've already spent too long on this. If nothing else, I've given you another weird data point! :)"
26,26,26,16,1,1,1,1,1,1,The top 3 are necessary for a majority and the 4th is also needed. The rest are filled in case my opponent leaves them empty.
26,5,5,5,6,7,26,0,0,0,"Most people will focus on high number, but castles 1-7 equal 28 points, enough to win. Realizing that someone may attempt to take castles 8-10 and castle 1, i redeployed troops to castle 1 to thwart that strategy. "
25,0,0,0,0,0,0,25,25,25,"The total points up for grabs is 55, and to win the war I need 28 points. I want to get 28 points by using the least number of castles, so I can put more soldiers in each castle and increase my odds of winning that castle. I can earn 28 points by winning castles 1, 8, 9, and 10. So I will put 25 soldiers each in castles 1, 8, 9, and 10 to maximize my odds of winning each of those castles simultaneously."
25,0,0,0,0,0,0,25,25,25,Submission #4. A variation of my third submission. Equally divided among just enough points to win. (Not convinced this will win either).
25,0,0,0,0,0,0,25,25,25,"There are 55 points up for grabs, so 28 are needed to win. Winning castles 1,8,9,10 are the fewest number of castles needed reach 28 points. Castle 1 is as important as castle 10 for getting to 28 points. "
25,0,0,0,0,0,0,25,25,25,"Since there are 55 available points, I only need to win 27.5 or more points to win any given battle. By maximizing my soldiers in the four castles that are worth 28 points combined, I maximize my chances of beating more evenly distributed enemies. "
23,1,1,1,1,2,2,23,23,23,"The ones and twos are mostly to pick up any undefended castles, while I hope to grab the highest castles to get me over 27.5. Have to admit I don't know much game theory, so it's mostly just a guess."
21,18,15,13,11,9,6,4,2,1,"On average 1.81 soldiers per point, with some slight weighting to the top castles and unweighting the lower castles."
21,0,0,0,0,0,0,26,26,27,"If you were to win castles 10, 9, 8, and 1 each time, you would win every matchup. I put all of my soldiers on those castles, with a few extra on the more valuable castles to beat out anyone with the same strategy"
20,12,13,13,14,14,14,0,0,0,Get to 28 by conquering the smallest towers
20,0,0,0,0,0,0,25,25,30,it put high power making it easy to win the castles with troops.
19,17,15,13,11,9,7,5,3,1,?
19,1,1,1,1,1,1,25,25,25,need 28 to win 
19,1,1,1,1,1,1,25,25,25,"The total number of points is 55 so you need 28 points to win the war.
The smallest combination of castles to win 28 points is 10,9,8,1 so to maximize your chances you should just split your army by 25 soldier each.
But this won't work because the other castles will be undefended and an enemy could easily put 90 soldier on Castle 10 and 1 soldier on each undefended castle winning the war.
So Castle 1 is defended by 19 soldier to be able to defended the rest of the castles with 1 soldier.
Running a simulation with a random number generator gives me a 98% chances of winning with this combination, althought it is sunday night and I might have made some fundamental mistake in the code"
18,18,2,18,18,18,2,2,2,2,To disrupt strategies that rely on lower value castles. 
18,16,14,12,10,8,6,4,2,1,
16,16,16,16,16,16,1,1,1,1,"Evenly distributing troops at 6 castles gives me a great chance to win a simple majority, and single troops at the remaining 4 gives me an auto win if my enemy leaves any empty. "
16,11,11,11,11,18,19,1,1,1,I can get 28 points out of 55 from the lower 7 castles so concentrate force there. Send a token soldier to the top castles in case someone tries a more extreme version of my strategy. Bias soldiers towards castles 6 and 7 because a 'aim at the higher castles' strategy is likely to still be interested in those. Send a few more to castle one because I could see a strategy of going for the top three castles and the lowest one.
15,14,14,14,14,14,15,0,0,0,"Target to win is 28 points. Concentrating deployment on highest-value castles means I need to capture 10, 9, 8 and 1 to reach target. Highest-value castles are likely to draw most troops by my opponent.  So I am going to focus on capturing enough castles from the lowest value upwards until I hit the target, which is castles #1-7 inclusive. Divide troops equally, with the spares focused on 1 (crucial to the 10-9-8-1 strategy set out above) & 7 (because it is the highest value of my targeted castles)."
15,13,13,14,14,14,14,1,1,1,"Contrarian strategy to go for bottom 7 (which sums to 28) rather than 10+9+8+another. Putting 1 on 8,9,10 to beat people who may use a similar strategy but forget that by getting these higher ones they win vs similar strategies. Higher on 1 to beat strategies that realize that 10+9+8+1=28 and guess 100/7=14 is lowest bound to beat on 1 (and assume people would pretty much always put the least amount of soldiers on castle 1). Even if using 10+9+8+other, unlikely to use higher than 12 or 13 on the lower castle because of the need to focus on top heavy. Strategies that assign soldiers based on castle value mostly beaten (unless using smoothing factor but this would lose to most top heavy strategies). Also beats 10 everywhere. Will see!"
15,7,8,5,5,11,10,11,10,18,People want 1 and 10 so I'm fortifying them
15,5,10,12,15,1,12,10,5,15,"Assuming opponents will want to send large forces at the beginning and end of their siege I fortified the first and last castles with relatively more troop. Same with the middle castle. The solitary soldier in castle six is a sacraficial lamb. Knowing I am bound to lose a castle or two, I am presuming that I can give up one castle to properly fortify more and eventually win the battles. "
15,3,7,10,15,19,25,2,2,2,"There are 55 total points available. Half-.5 of those are up for grabs in just the top three castles. So a strategy to just hold the top 3 castles would fall barely short of success; a strategy of holding the top 3 castles plus Castle 1 would barely succeed, and Castle 1 might be viewed as the lowest-cost place to pick up that one needed point. Assuming that most people will gravitate towards this upper-tier strategy, I chose to do something that would win against it."
15,1,1,1,2,2,3,25,25,25,"Need 28 pts to win a war.  I assumed there are no loser points in war.  I considered various strategies, even dist, weighted dist, lower focus to 28 pts, upper focus to 28 pts, and a few variants before settling on the above which seemed to have the best results."
15,0,0,0,0,0,0,17,26,42,Submission #5. I guess I have the second most confidence in this (of my 6 submissions).  Defending just enough points/castles to win and dividing them unequally in (probably vain) hopes that I can win. 
14,14,14,14,14,15,15,0,0,0,"To win a war I need 28 victory points (round up half of the total number of available points).  Figure most people are going to try to target the high value targets (8,9,10) which together make 27 points.  So if I can capture the rest, I win."
14,14,14,14,14,15,15,0,0,0,"With limited math skills, my basic thinking is that in this 1v1 scenario I just need 28 points to be victorious since the total number of points is 55. 1+2, +7 = 28, where 8+9+10 = 27. So I'm able to capture the first 7 castles while leaving my opponent to deploy most of his troops on the higher value castles (because who wouldn't normally want the highest value castle?) then I have the highest chance of succeeding and capturing 28 points. Hope I won. PS I see that double half-zip."
14,14,14,14,14,15,15,0,0,0,"You need 28 points (a majority of 55) to win.  I am guessing that it will be easier to do that if I focus all of my troops on 1-7 and none on 8 through 10, because I think the majority of people will overvalue those.   "
14,14,14,14,14,14,16,0,0,0,"There are 55 points total, and I need 28 to win. Most people will concentrate on the higher numbers. "
14,5,6,8,10,12,14,10,10,11,"Long story short, a ton of meta-gaming. I'm an avid gamer, so analyzing strategies that are in 'meta' is something I do often. I developed 6 (really 3 archetypes) solid strategies, which would each win in a random environment. The 'uber meta' strategy that won against every strong, solid strategy whilst still being able to win an 'average' game against randoms was this one, developed out of my ""Empty Castle"" strategy. I know this may sound pretentious, but if I win, I would very much like to show you my ""strategy notes"". They're too complicated to describe in words, but pictures could give you a better idea. Cheers!"
13,13,13,14,14,15,15,1,1,1,Need 28 points to win = win castles 1 through 7. Spread troops close to evenly among those castles and put 1 troop each on 8-10 in case opponent is using similar strategy. 
13,13,13,13,13,13,13,3,3,3,"I chose this troop deployment because 28 victory points are needed to win. Therefore, if I won all of the lowest 7 castles, then I'd win, but I put some on the last three in case someone else highly concentrates their troops in other places."
13,7,17,8,11,6,13,7,9,9,Had some wars on excel tempted to do a straight 10 per castle though.
13,7,10,13,16,19,22,0,0,0,"forfeit on 8,9,10,   overweight 1, "
13,3,13,31,24,11,1,1,2,1,I figured people would go after the later castles and keep their deployments balanced
13,0,0,0,0,0,0,29,29,29,"Seeing as there are only 55 total points available, you only need 28 victory points to win. The ""easiest"" way to do this (in terms of total number of castles won) is Castles 1, 8, 9 and 10. 
I then split the number of soldiers such that the ratio of soldiers at castles 8 to 9 to 10 is 1:1:1 and the number of soldiers at castle 1 is greater than 10. 
This strategy will beat anyone who splits evenly between the 10 castles, and (I'm hoping) will beat a decent number of people who go for the same four castles.
An example strategy this would lose to is is someone split all 100 of their troops between e.g. Castles 9 & 10. I decided not to employ a similar strategy since I think more people will try something similar to mine rather than something somewhat counter-intuitive like betting all their troops on only two castles (although this isn't really based on any evidence)."
13,0,0,0,0,0,0,29,29,29,Focus all troops on the fewest number of castles that would win the minimum 28 points necessary to win. 
12,12,12,12,15,17,20,0,0,0,I gave up 3 castles and tried to win 7. My hope was that others would try to win the high point bases and I would therefore be able to steal the bottom bases and the win.
12,12,12,12,13,13,26,0,0,0,"I assumed that the most popular strategies would be a distribution close to 10 everywhere, a distribution close to putting a number of solders in each castle equal to (100 * castle # /55) and strategies which only attack castles 7 through 10. This strategy requires that I win castles 1 through 7 so each castle is worth the same to me, except I need to make sure I steal castle 7 from the people only going for 7-10 (and one of the variations there is to play 25 soldiers across the board)."
12,12,12,12,13,13,13,13,0,0,Trying to make them waste troops on 9-10
12,12,12,12,12,12,26,1,1,0,Beat the 10x10 strategy or any that over values the last 3 castles. If you win 7 you have 28/55 pts. 
12,12,12,12,12,12,12,12,2,2,I does not matter if I lose more than one castle from 1-8.  None are more important than the other.  7/8 in any combnation wins here.  I choose to use 2 in castle 9 and 10 to pick off empty castles or somebody that chooses to put 1 in there.
12,12,8,8,10,10,10,10,10,10,Just enough to trip up a full 10-10 average deployment (maybe? :) )
12,12,2,4,4,11,19,10,18,8,"Used a random number formula which summed to 100.  Recalculated a few times to get some heavier numbers near the castles 6-9, since I assumed most people would heavily load castle 10."
11,12,13,14,15,17,18,0,0,0,"While last few castle have most points, they are also more heavily defended. I am shifting my troops down the order, and give up the big castles. Smaller castles are easily overwhelmed by the extra troops I placed."
11,12,13,14,15,17,18,0,0,0,"There are 55 total points available, so 28 points are necessary to win. I figure a lot of people will focus on winning the high value castles, so I will focus on winning enough low value castles. I didn't think about this strategy very long, but I hope I beat somebody."
11,12,13,14,15,16,19,0,0,0,"Out of 55 points you want 28. You want to put more troops in higher valued castles, however by using only 1-7 (and conceding the highest 3 castles) I can reach the necessary 28 points. I divided it evenly between the castles with a singly increasing importance on each higher castle. The added remainder went to the highest sought after castle.  "
11,12,13,14,15,16,19,0,0,0,"sum of points for winning castles 1-7 is greater than points for 8-10.  i'll let others win those.  need to be able to beat people that send 1/10 to every castle, so castle #1 needs at least 11 soldiers.  increased by one up to  castle #6, then send the rest (19 soldiers) to castle #7"
11,12,13,14,15,16,17,0,0,2,We only need to get 28 of the 55 victory points to win. Castles 1-7 deliver that for us. Let others focus on trying to get Castles 8-10. Put a leftover 2 on 10 to counter anybody else trying a similar strategy of ignoring Castle 10.
11,11,12,14,14,18,20,0,0,0,"The goal is not to win all the castles; the goal is to win the majority of the 55 points possible. I assume that more people will focus on winning the higher point values (7, 8, 9, 10) so I will take the opposite strategy. I can win 28 points if I win castles 1-7, so I split my troops among those castles, more or less equally. I can assume that, if people try a 1/8/9/10 strategy, they will not weight 1 as highly."
11,11,12,13,14,16,20,1,1,1,"Beats strategies that choose 10 equally, prioritizes the first 7 castles since 1+2+3+...+7=28 > 55-28 = 27.

1 bids on Castles 8-10 to deter people from not putting any troops in those areas at all."
11,11,11,13,16,18,20,0,0,0,"I chose to not contest 8, 9, and 10 which may be attractive to go after since they are the highest point value castles, and save my troops for going after castles 1-7. If you are able to win castles 1-7, you automatically will win with a score of 28-27. I chose to at least put greater than 10 soldiers at each castle that I wanted to contend for, such that I would win if my opponent distributes evenly (i.e. 10 soldiers at each castle), that I would still win castles 1-7. I then increased my troop levels for castles 4-7 such that all 100 soldiers were distributed, with castle 7 receiving my most troops, in case my opponent tries to stack all or most of their soldiers at the higher value castles. Additionally, if my opponent also puts 0 troops at castles 8-10, we would split points and I would have a chance for an even larger number of victory points."
11,11,11,12,12,13,27,1,1,1,"Half of available points is 27.5. If I win castles 1-7, that is 28. If someone distributed evenly I want more than 10 on the lower castles. If they try for the top 4 I want to have more than 25 on castle 7. If they try for a relative expected value strategy I want more than 12 on castle 6. If they try my strategy (bottom 7) I want to steal castles 8-10 with the minimum number of troops. "
11,11,11,11,13,17,26,0,0,0,"Castles 1-7 are enough to get the majority of the points. This allotment defends against putting 10 in every castle and putting 25 in the top 4 castles, and should beat many strategies that focus on seriously competing for the top castles."
11,11,11,11,12,16,22,2,2,2,"I did not put a focus on the top 3 numbers because if I captured the bottom 7 I would win, and I figured others would place a large amount in the 10-9-8 castles. I would have no real ability to guess the arrangement others used for the big 3, so I went small. I believed I needed to be able to beat a 10 soldier in every castle lineup, so I needed at least 11 soldiers in every castle numbered 1-7. I decided the 7 castle was likely my most important, so I wanted to creep the people creeping 20. With the 6 I creeped 15, and the 5 I creeped people creeping 10. All of the other low castles I creeped 10. In order to beat others running a low castle strategy I creeped people putting 1 in the 10-9-8 castles."
11,11,11,11,12,12,26,2,2,2,"To win each round, you only need to win 28 points. Winning any more than 28 points doesn't do you any good - a win is a win, whether by 1 point or 10 points. I identified what I believe will be popular strategies, and developed one that can beat them. I suspect the most popular strategies will be the Even All Strategy, the Top 4 Strategy, and the Bottom 7 Strategy. I also believe some players will employ versions of the Marginal Strategy, which I employ here. I am playing what I call the Marginal-Low Strategy. This troop deployment is susceptible to a variation of the Even Strategy that evenly distributes forces to a smaller number of castles, rather than 10 across the board."
11,11,11,11,12,12,26,0,0,6,"I basically tried to come up with a solution that would beat the most common solutions I could think of.  Being that I had no idea what others would submit, seemed like the best thing to do."
11,11,11,11,11,22,23,0,0,0,"Over half the points are in 1 - 7 (28) vs (27) in 8, 9, 10. This will beat an even spread of 10 x 10, or 5 x 20 in the top 5 castles. This should beat most strategies that put most points into the top 3 castles. The only strategy it would lose to would be a 6 -10 castle strategy that weighted its troops to castles 6 and 7, not 9 and 10 - which seems an unlikely strategy to take and would lose to the commonsense strategy of putting more troops in 8, 9 and 10.  "
11,11,11,11,11,21,21,1,1,1,"I tested it out against all of the strats I thought people might use, and this one won against all of them"
11,11,11,11,11,21,21,1,1,1,"We started by examining several strategies that we expect our opponents will be likely to employ - including: (1)equal distribution strategy across all castles, (2) gradient strategy with highest deployment at castle 10, (3) sacrificing castle 10, with gradient starting at castle 9, (4)sacrificing 10 and 9, with gradient starting at castle 8, (5)sacrificing 10, 9 and 8 with gradient starting at castle 7, (6)equal distribution across all even numbered castles, (7)equal distribution across all castles from 1-7.
We then distributed our soldiers in a fashion that would defeat all of these seemingly reasonable strategies.

Bring it on arch-enemies!"
11,11,11,11,11,21,21,1,1,1,"cuz why not, lol"
11,11,11,11,11,20,25,0,0,0,"I'm attempting to maximize my odds of getting 28 points out of a possible 55, this guaranteeing victory. I'm ceding 8-9-10 thinking most people will throw all their troops that way. I'm also thinking there will be enough people who assign 10 troops per castle, which is why I put 11 troops on the lower numbers."
11,11,11,11,11,19,26,0,0,0,"I enjoyed this week??s Riddler. I attacked it, not mathematically, but by brute force and trial n?? error. I learned that the best strategy would involve trying to win a few key battles (i.e. not all of them), loading to ensure victories in those battles, and that it would entail barely winning in the end; i.e. a small margin of victory. 

My first thought was to look at ways to lock up the highest-value castles. Winning the battles for the top 3 castles is 27 points, only 1 short of victory, so my approach involved throwing a lot of soldiers at the top 3, a chunk at a lower-value one, and deploying 1 soldier at the remaining ones (to win battles against zero soldiers). An example of this approach is 0-2-1-1-1-1-1-30-31-32. This wins against many strategies but fails against a simple one of 10-10-10-10-10-10-10-10-10-10. Loading up on one lower-value castle to 11 (to defeat that strategy) leads to too few soldiers at the higher-value castles.

Then I thought of the opposite approach; i.e. concede the battles for the 3 higher-value castles and try to win the remaining 7 (which would yield 28 points, and a win). The best approach I found was 11-11-11-11-11-19-26-0-0-0-. The 26 is necessary to defeat a strategy of deploying ?_ of one??s soldiers (i.e. 25) to each to the top 4 castles, the 11 is to beat the 10x10 strategy, and assigning the remaining 8 soldiers to the 5th highest castle.

This strategy works against almost every strategies, especially the ones that many people likely would choose. It fails against strategies involving loading up on the mid-value castles; e.g. 0-0-1-4-11-20-25-20-15-4. However, as those strategies lose to many other ones I thought people would not choose them."
11,11,11,11,11,19,21,1,2,2,"Made up some simple strategies (10 each, top 3 heavy, bottom 7 only, proportional to points) and this beat all my toy scenarios. Then among those playing for the bottom 7 I wanted to sometimes steal one of the big castles if they leave them wide open, and also weight toward the 6th and 7th castle to hopefully win those against similarly minded players and also win from some people doing something proportional to points but giving up on a few bottom categories or going after 5 columns 20 each."
11,11,11,11,11,16,25,1,1,2,"The name of the game is getting to 28 Victory Points. If you win Castles 1-7, you reach 28, and the remaining castles are useless. Therefore, it makes sense to load heavily on 1-7 while virtually ignoring 8-10. I chose to throw a point or two on the big ones, just in case someone else uses the same strategy, but chooses zero for any of the top three. Also, using this strategy requires more than 10 on every value 1-7, because otherwise it would fail against an even distribution of 10 per castle on tie breakers."
11,11,11,11,11,14,25,2,2,2,"Winning 1-7 or 7-10 wins enough points to win the war.  This makes 7 the most important castle.  I chose the low points strategy, assuming it would be the least used, and that people that go high, will dump more points on 10.  So the plurality of my points go to castle 7 to win it.  I put 2 points on 8-10 to win any zeros and ones (for the people throwing a token army at the high numbers).  "
11,11,11,11,11,11,11,11,12,0,"I figure everybody will divert too many for castle 10, so I divided up for the remaining"
11,11,11,11,11,11,11,11,11,1,"Hopefully people divide equally, and this maximizes my chances against such players. If they do, I win 9 of 10, and lose castle 10"
11,11,11,11,11,11,11,11,11,1,Assume everyone else will over-allocate to castle 10. Sacrifice and make up points elsewhere.
11,11,11,11,11,11,11,11,11,1,
11,11,11,11,11,11,11,11,10,1,"Overloaded each castle except for 9 & 10. Put one in 10 incase someone else avoided it as a high risk castle, and took it from the 9 as it's a slightly higher risk castle."
11,2,11,11,11,2,12,36,2,2,"I assumed a whole bunch of people smarter than me were spending hours on the mathematically best way to deploy your troops, so I went with an opposite approach: Randomly guess which castles to deploy to, while aiming to gain 28/55 possible victory points. I chose castles 1,3,4,5,7,8 and weighted more heavily towards to castles worth more points. I chose 11 as a minimum so that I couldn't easily lose to someone who just put 10 troops in each castle, and 36 in castle 8 so that someone with the 1-8-9-10 strategy also wouldn't win. I also killed about 10 minutes at work, so I'm pretty happy."
11,1,12,1,15,1,19,19,20,1,Trying to maximize expected value knowing my opponent will be doing the same thing. 
11,1,1,1,1,1,1,26,26,31,Go For 28 points
11,1,1,1,1,1,1,25,27,31,"There's 55 points, so you need 28 to win.  10+9+8+1 = 28; that's the fewest number of wins.  Guard against people splitting their troops 10 ways by sending 11 to Castle 1, and don't leave anything uncontested by not sending any.  That's 11 + 6 = 17, with 83 to spread out between Castles 8 - 10.  I didn't overthink this."
11,1,1,1,1,1,1,20,29,34,"There are 55 total points available, so you need 28 points to win, which is why I focused on castles 1, 8, 9, and 10. Winning those four castles gets me exactly 28 points. Assuming an average of 10 soldiers per castle, placing the amount of troops I put in each of these four castles should hopefully let me prevail more often than not and get the 28 necessary points to win the Battle Royale and rule Riddler Nation."
11,0,1,1,1,1,1,25,28,31,"Assume enemy will try to be clever and will have assumed that I am targetting large castles. He will have allocated his troops to win lower castles. So I try to win the point total by reallocating to win castles 8, 9, 10, and 1. In case any of the middle castles (3-7) are ignored, send 1 soldier to prevent a point split and steal the win."
11,0,0,2,3,3,3,26,26,26,"The easiest way to win is to win {1, 8, 9, 10} for 28 v 27. 
The strategy needs to counter:
[*] Strategy who tries to win {7,8,9,10} and goes all-25: This means that {8,9,10} must have at least 26 soldiers.
[*] Strategy who splits 10 soldier to all: This means that {1} must have at least 26 soldiers.
This means we have
1 : 11
8 : 26
9 : 26
10 : 26
remaining 11 soldiers
The only enemy for this strategy would be strategy who goes kamikaze and play for {9,10} and goes split-50. The remaining 11 soldiers is split for {4,5,6,7} to make up for the 19 points loss from the kamikaze play."
11,0,0,0,0,0,0,29,29,31,
10,11,12,13,15,17,22,0,0,0,I think people may gravitate toward locking down high numbers.
10,10,20,30,5,5,5,5,5,5,I can capture the lower numbers while everyone else is fighting over the higher numbers. 
10,10,11,12,12,13,26,2,2,2,go for the lower castles to get 28 points
10,10,10,15,15,20,20,0,0,0,"I figure castles 1-3 will be lightly deployed while 4-7 will be slightly more defended. But I expect most will focus on the big targets. If I can win castles 1-7 and ""sacrifice"" 8-10, they will only have 27 points. I will have 28 points and still win the war."
10,10,10,11,11,12,12,12,12,0,"In most scenarios, it would beat what I'd plan if I was being lazy."
10,10,10,10,15,20,25,0,0,0,"I know I need 28 pts to win.  Taking castles 1-7 gives me 28 pts.  Many people will likely put a large # of troops on one or more of the bigger point value castles. Of course, many people will think of this and do the opposite, making my theory kaput. "
10,10,10,10,15,20,25,0,0,0,"If you win castles 1-7 you win the game.  This strategy hopes the enemy will waste all or most of its army's to win 8-10, and not be prepared to win any one of the remaining castles."
10,10,10,10,10,10,10,10,10,10,"If my opponent also places ten at each castle, we tie. At worst, I win half the match-ups, and we tie again. But if my opponent places more than ten at one or more castles, chances are I'll win more points. "
10,10,10,10,10,10,10,10,10,10,Uniform Distribution
10,10,10,10,10,10,10,10,10,10,
10,10,10,10,10,10,10,10,10,10,Maximin
10,10,10,10,10,10,10,10,10,10,Guaranteed to win at most half the points per battle (55) to at least Castles 1-5  (15)
10,10,10,10,10,10,10,10,10,10,Just a wild guess at a strategy that might stand up against the widest array of possible configurations.
10,10,10,10,10,10,10,10,10,10,Not sure 
10,10,7,5,7,7,16,11,13,14,"set.seed(154)
poo <- sample(1:10, 100, replace = T)
table(poo)"
10,8,8,13,8,20,30,1,1,1,"There are a maximum of 55 Points available, so 28 is a Winning score. My strategy is to win the first 7 castles to get 28 points, hoping my opponents over commit solders to the last 3 castles. I have also overcommitted to castle 1 as Castle 1,8,9,10 is a winning strategy same applies to castle 4 as 4,7,8,9 is a winning combination. "
10,1,1,1,1,1,1,27,28,29,The plan is to get to the 28 points needed with as few castles as possible while also leaving a guard against other strategies that assign zero soldiers to some castles.
10,0,0,0,0,0,0,30,30,30,"If I win castles 1,8,9,10 I will receive 28 points and my opponent will only receive 27.  This is the least number of castles that I need to win in order to beat my opponent.  Assuming that I need to win all 4 to have a chance and that my opponent will put very little into winning a single point from castle 1, I will only deploy 10 soldiers.  Winning castle 1 is a lynchpin though and I need to win it or my theory fails.  All other castles are very important to me and I am able to put 30 soldiers in castles 8, 9, and 10 because castles 2-7 mean nothing to me.  I can lose them all and still win if I win 1,8,9,10. "
10,0,0,0,0,0,0,30,30,30,Load up the soldiers on the minimum castles needed to win
10,0,0,0,0,0,0,30,30,30,Tried to choose the fewest number of castles (and in the case of #1' the least likely to be attacked) to attack that would give me a majority of the points.
10,0,0,0,0,0,0,30,30,30,"Deploying hopefully overwhelming force at castles 8 through 10, and a token force to capture 1. It doesn't allow any room for failure, but hopefully will be strong enough at the one point to ensure victory."
9,12,13,14,15,16,17,1,1,2,"I attempt to concede castles 8-10 to win 1-7 in order to scrape by with a total win. I moved one extra soldier from 1 to 10 in case someone puts 1 in the last 3 castles, but otherwise uses the same strategy."
9,11,11,11,11,11,11,23,1,1,"I tried a ton of different variants of a number of strategies, which I ran against each other using python code. I came to a point of realization that any strategy could be beaten by another strategy, but that it would probably be advantageous to choose a strategy which eschewed the top two values completely and beefed up where others would not think to invest but would add up assuming over-investment in the highest values. The particular distribution I chose, while simple, performed the best overall against top-heavy distributions."
9,11,9,11,9,11,9,11,9,11,I thought I'd have the most wins by splitting up my troops almost evenly.
9,10,10,10,10,10,10,10,10,11,"Wanted to get the most possible free points from people who are just skipping out on castles, so I figure just send troops everywhere. You miss all the shots you don't take. Slightly altered the troop deployment for Castle 1 and Castle 10 so as to beat anyone else using this strategy."
9,9,9,11,9,12,9,11,10,11,"The goal of a battle is to get to 28. For simplicity's sake, I ignored ties (likely a fatal omission). So I considered all 36 partitions of 28 into distinct parts less than 11. Each of these partitions represents a winning coalition of castles that will get me to 28. Then I counted how often each castle was in a winning coalition and used that to get a percentage of my troops to send to that castle. After truncating some percentages I had one trooper left over, so I put him in castle 6, which is the highest value part that appears the most. In hindsight this leads to a very uniform distribution of soldiers, which makes me question my methodology, but let's see how it does."
9,9,9,9,9,11,11,11,11,11,I just want to beat the person that deploys them evenly to every castle!
8,10,12,14,16,17,19,2,1,1,"My plan is to try to pick up all of the lesser value castles that others have hopefully ignored. If I can get castles 1-7, then that is enough to win. I've reserved 1-2 extra troops for the remaining castles just in case the opponent's strategy involved abandoning one of those altogether."
8,10,11,11,12,13,14,6,7,8,Somewhat even distribution for people who include lots of low numbers and some slightly higher ones for the people who include near totally even distribution. 
8,9,12,14,16,18,20,1,1,1,"For the most part, I gave up on the top 3 value castles (total points 27), to focus on the bottom 7 value castles (total points 28), as I thought most people would commit a lot of resources to the top value castles, and I could win all of the lesser battles.  However, I left 1 army at each of the top value castles to combat a similar strategy to mine, except where they left the top castles completely undefended."
8,9,10,12,15,18,25,1,1,1,I figured people would go for the larger numbers.
8,9,10,10,15,22,26,0,0,0,"You do not need the castles worth 10, 9 and 8 to win the battle, so long as you have all the other castles. Therefore why waste men on the castles which are most likely to be attacked."
8,8,10,10,10,10,11,11,11,11,"Strategy A:  Concentrate your forces as much as possible, taking more than half the victory points while holding the minimum number of forts required. (forts 10, 9, 8, x).  Likely choose fort 1 for x since it's sufficient and unlikely to be hotly contested.

Strategy B:  Beat A by focusing on forts 8, 7 ,6, 5, y.  Where y is not 9, 10, or x.  Since ""Strat A"" will likely put 32 or fewer soldiers in each of forts 8, 9, and 10, Strat B puts 33 soldiers in fort 8.  It then picks up 20+ victory points from forts 7, 6, 5, y.  Likely to choose fort 2 for y since it's sufficient and probably not hotly contested.

My chosen strategy (C: fully distributed) is likely to beat simple variants of A and B (as well as many other random or semi-random distributions).  Since I expect those strategies to be most common, that's what I'm going with.

Basically, this puzzle is much like the Riddler Express puzzle; both come down to the player's estimation of other player's strategies."
8,8,9,9,13,20,30,1,1,1,"with a total of 55 points available, i conceded the higher level castles and focused on the smaller castles to win the majority(spoiler: like how the electoral college went lol)"
8,8,8,8,8,25,30,3,1,1,I thought 7 was most important
8,2,11,12,17,2,21,21,2,2,"55 points are available, common strategies to get 28 may involve attempting to getting a few high scoring or many low scoring castles. 8,7,5,4,3,1 gets 28, with 2 soldiers minimum to each castle in case of uncontested/1 soldier chosen by an opponent, and avoids relying on the highest castles or too many castles"
8,2,4,11,8,14,13,9,14,17,"https://goo.gl/qwoylN

wrote this code to randomly generate 1000 'setups'
then battled each setup against eachother and only returned the 'setup' with the most wins

a general strategy is to try and accumulate 28 points total to guarantee yourself a win. This can be done with a minimum of 4 wins (on the 10,9,8 and 1 point castles)."
8,0,1,1,2,5,11,24,10,38,evolutionary programming
7,13,14,15,16,17,18,0,0,0,Split relatively evenly between less contested castles
7,11,0,14,16,0,23,0,29,0,I want a winning coalition of 28.
7,10,12,14,16,18,20,1,1,1,"I am ""punting"" the top three castles, hoping my opponents waste troops.  Therefore, I can win the bottom seven castles to secure victory (28 pts).  "
7,9,11,14,16,19,24,0,0,0,"I thought maybe most people would go for the big money, so if I can win the 28 points for the seven smallest castles, that would beat most strategies."
7,8,11,14,17,20,23,0,0,0,"I expect that many others will choose to allocate troops to the highest values (8/9/10). If I can win with just Castles 1-7, it doesn't make sense for me to devote any troops to 8-10 since these troops will usually be completely wasted. The weakness of this strategy is that it gives me an extremely narrow path to victory -- I need all seven castles to win (in the absence of ties from 8-10). Therefore, each is essentially of equal importance to me (losing either 1 or 7 cripples my chances). My troop allocation, then, is more about where other players will choose to allocate troops. Most players will likely choose a proportional strategy, where higher values also have higher troop allocations, so I also have a proportional allocation. After distributing troops proportionally, I am left with two extras. My suspicion is that some players will try to allocate a couple extra troops to 1, thinking that it's an easy way to pick up a win. Because I need castle 1 to win, I have allocated these extra troops to castle 1 to ensure a victory there."
7,8,10,12,15,20,25,1,1,1,"I figured everyone would fight over the top three castles, so I'm making a risky gambit by ignoring them and hoping to sweep the bottom 7."
7,8,9,11,12,13,14,15,5,6,I know that everyone would likely (on average) assign increasing numbers based on the value of the castle.  My goal is to be out of 'phase' with those values in order to be 'slightly' higher than the most castles that I can...
7,8,9,10,11,12,13,14,16,0,"I decided to sacrifice points at the highest value castle (10 point castle) and provide myself with a higher probability of winning the remaining castles. I then deployed troops in descending order from castle nine down to castle one.

Jon Snow had to sacrifice Castle Black (or at least it's tenant's ideals) to (hopefully) regain the North, so I was inspired to do the same!"
7,7,15,15,15,16,16,3,3,3,It is the opposite of my other strategy
7,1,1,1,16,19,23,1,30,1,"Focused on 5 castles (1,5,6,7,9) that would yield 28 victory points - just enough to win. No hard and fast rule on choosing castles - just what I thought would be an interesting way of achieving it - didn't want to be too overweight in high value or low value castles so chose a mix. Sent 1 troop to remaining castles (2,3,4,8,10) in case opponent had similar strategy / didn't send troops. Split remainder of troops roughly equivalent to weight of castle in terms of victory points (approximately 3 troops per VP)."
7,0,14,15,10,17,16,18,2,1,STRATEGY!  Banking on the little guys!
7,0,0,0,0,0,0,31,31,31,28 or bust.
7,0,0,0,0,0,0,31,31,31,"There are 55 points total to be won between the 2 warlords if all castles are fought for, so whoever gets 28 or more wins in that case.  In that case there are 14 ways to get at least 28 points, by winning one of the following specific groups of castles: {10,9,8,7}, {10,9,8,6}, {10,9,8,5}, {10,9,8,4}, {10,9,8,3}, {10,9,8,2}, {10,9,8,1}, {9,8,7,6}, {9,8,7,5}, {9,8,7,4}, {8,7,6,5,4}, {8,7,6,5,3}, {8,7,6,5,2}, {7,6,5,4,3,2,1}.  I would like to try to win the fewest number of castles yielding at least 28 points and including a castle that fewer warlords would desire if possible so I can win it with a light deployment and concentrate in the others.  From the above, it appears that a 4-castle group of {10,9,8,1} satisfies that, so those are my targets and I have concentrated the soldiers in the higher values castles as desired."
7,0,0,0,0,0,0,31,31,31,Well i was in the armed forces for about 27 years soooooo i think i know what I'm talking about pfffff
7,0,0,0,0,0,0,23,30,40,"There are a total of 55 victory points available, so 28 are needed to win each war. Winning is not necessarily about getting the most victory points -- it's about getting to 28 victory points as often as possible. Thus I dumped almost of my troops in the 8,9, and 10 victory point castles, since winning those three is a total of 27 victory points. Unfortunately, I needed one more victory point, so I put 7 in the 1 victory point castle, hoping that it would be virtually ignored by most people. If one were to distribute troops to castles proportional to their victory points, only (1*(100/55))= 1.818 (which rounds to 2) would be sent there, so I hoped 7 would be enough to take care of that. "
6,12,9,15,15,0,0,21,21,1,Picked favorite numbers.
6,8,11,14,16,18,21,2,2,2,Concentrate on the smaller ones.
6,8,11,11,16,21,21,2,2,2,Focus on the mid-valued castles
6,8,10,14,18,20,24,0,0,0,"win all 7 and lower, sum = 28, higher than 8+9+10."
6,8,10,12,14,24,26,0,0,0,Guess from the bottom assuming most will picking from the top
6,8,9,16,18,21,22,0,0,0,
6,7,9,12,16,21,29,0,0,0,Triage - concentrate forces to get minimum points to win. Abandoned castles worth most points in hopes opponents would concentrate their forces there and I sneakily win with the lesser valued ones.
6,7,8,10,10,21,38,0,0,0,"There's two basic  strategies: quantity(1-7) and quality(7-10). In either case, the 7-point castle is the most important one. I spelled out here a quantity strategy. To do so, I applied a Benford's law curve with the highest value on 7. Why Benford's? No reason, I just felt that a curve like that was a good representation of the quantity strategy."
6,7,8,9,10,13,14,15,18,0,"Concede the big prize, deploy those available soldiers to hopefully win the rest!"
6,7,8,9,10,11,12,13,14,10,Worth plus 5 except for the 10 point castle
6,7,8,9,10,10,11,12,13,14,Jdnd
6,6,10,12,14,15,16,17,2,2,Focus on a large group of middle value targets.
6,6,8,15,20,15,10,8,6,6,Concentrate in the middle
6,6,7,11,16,21,26,3,3,1,"Since the high value castles I'm intending to fight somewhere else. By ignoring the place where (I hope) most players are spending their resources, I hope I have the forces to win across the board on 1 through 7. Overspending slightly on the low numbers to counter strategies which try to spend high on one or two low numbers to pick up an edge.  Oh, and playing 6's and 1's  to come in just over anyone sending forces in round numbers."
6,6,7,8,10,25,35,1,1,1,The ancient chariot race story
6,6,6,32,32,7,11,0,0,0,Win more towers (with lower points each) so that total points compensate for losing highest point towers. Towers 1-7 have 28 of 55 points. Towers 1-5 have 15 of 28 total points among towers 1-7.
6,6,6,11,20,21,21,3,3,3,Let people fight for the big ones. 
6,6,6,6,6,7,15,16,16,16,"This deployment is what I think has the highest number of troops that need to be correctly to defeat it, with 56 out of the 100. Additionally, although two other deployments also have that minimum (one troop from 8->5, and from 6->7), this has the fewest possible ways of reaching that number, with two (as opposed to four for the 8->5 and six for 6->7). This is the maximin for the problem, so that's why I'm choosing this for my deployment.

The minimum deployment to beat this is 6 to either castle 4 or 5, 16 to castle 7, and 17 to castles 9 and 10, which gets 2+7+9+10=28 or 2.5+7+9+10=28.5"
6,2,2,15,19,23,30,1,1,1,"Only need 23 to win. Everyone will go for 8-10, so I will go strong for 4,5,6,7 and #1 to get my 23.
"
6,1,8,12,10,1,30,30,1,1,"attempt to win the minimum number of points concentrating on lower values that should be less competitive, leaving 1 troop to pick up any undefended castles"
6,1,1,1,1,1,1,21,21,46,Because the optimal strategy should be to win with the fewest castles
6,0,0,0,0,0,0,26,27,41,
5,15,15,15,15,15,20,0,0,0,"I figured that if I won the bottom 7 castles, the total points (28) would outweigh the top three opportunities for points from Castles 8-10 (27). However, knowing the slim difference, I thought that breaking up the 100 soldiers to each Castle 1-7 would be the best strategy, knowing that if my opponent won Castles 8-10 they couldn't take any other Castle or else I would lose. "
5,12,17,13,13,11,13,5,4,7,"monte carlo simulation of 100,000 placements."
5,10,20,10,5,5,20,10,5,10,Expecting to crush people who put 0 a lot or overdo #10
5,10,10,15,15,20,25,0,0,0,"Assuming most people will try to take at least one of the high-value castles, I send disproportionately high numbers to the lower value castles in an attempt to sweep all 7, and win the war 28-27"
5,9,15,14,16,16,17,0,0,8,Winning all the castles from 1 through 7 gives you a victory no matter if you win or lose the remaining three castles. So using that I weighed my army heavily on the lower portion figuring most people's gut instinct would be to distribute their troops with powers relative to the VP for a castle. I gambled on people giving up on the lower castles so taking those for free points and putting a good number of troops in the middle range where I expected the most resistance to the strategy.
5,9,12,14,16,19,22,1,1,1,"I need to win 28 points- there is no prize for winning all the points. So- rather than go after Castle 9/10 which will be heavily contested, I'll aim for capturing 1 through 7 to earn 28 points. I shouldn't completely neglect the higher castles- if they're undefended I might as well capture them (because they must've stationed their troops elsewhere!)"
5,8,12,15,18,22,5,5,5,5,"This beats the vast majority of strategies that do the following:
focus attention on a subset of castles totaling >=28 points, assigning soldiers proportional to points for castles in the subset, and zero elsewhere."
5,8,8,10,13,16,37,1,1,1,"There's two main strategies: 7 8 9 10, and 1 2 3 4 5 6 7. Castle seven is, then, the most important castle. My strategy seeks to prey upon the first strategy, with a miser's troop in each expensive castle, to try to prey upon the second strategy."
5,7,11,14,17,21,25,0,0,0,"28 points to win. ~3.5 troops per point. {10, 1} are focal, as is strategy: max points-per-castle (ie assign in descending order). Current distribution averages troops-per-point over castles least focal, with kinks at 1, 7 to account for bias towards those strategies. "
5,7,10,14,18,21,25,0,0,0,"I gave up (zero troops) castles 8, 9, and 10 and their total score (27).  I then crossed my fingers and hoped that by heavily weighting the bottom end, I could sweep castles 1 through 7 and receive 28 points.

Now that I look at the logic, I see other alternatives.  

meh. I'll stick with this one."
5,7,9,12,15,22,30,0,0,0,"calculated that as winning castles 1-7 outscores winning 8-10, so I chose to concede castles 8-10 assuming that is where the majority of people chose to put their soldiers allowing me to put more troops where i assume less people put there soldiers"
5,7,9,12,13,14,18,20,1,1,"Ideally, I want to beat my opponent by as little as possible in as many castles as possible.  The obvious plan is to assign many troops to high-value castles, and fewer and fewer for each castle down the road.

I plan to surrender the 10 and 9 pointer, only assigning 1 troop to each on the chance that someone assigns zero.  From there, I have a decreasing amount of troops from castle 8 to castle 1.  Hopefully I will massively lose the battle for 9 and 10, but win the war for the remaining castles."
5,7,9,11,15,21,25,2,2,3,"Winning the lower 7 gives you more than half the points, so the top 3 values are largely ignored save for a scouting force of 2-3 to prevent a lone scout of the opponent from stealing. Then starting at 7 we deploy in force and curve down from there."
5,7,9,11,12,15,17,19,1,4,Just tried to defeat what I thought might be a few popular strategies.
5,7,8,14,18,23,25,0,0,0,Assuming more people will try and fight for the larger point value castles. By not dedicating any resources to those top 3 point value castles I hope to win the remaining 7 and win each battle with a score of 28-27
5,7,5,7,12,11,15,13,13,12,"I wrote an R script to generate random arrangements of troops, and then I compared them against each other.  My program ran very slowly, and this was the best arrangement of troops it came up with."
5,6,10,10,10,10,15,30,2,2,Focus on getting more low-value castles without totally ceding 9 & 10
5,6,7,8,12,13,14,15,20,0,"I sacrificed Castle 10, predicting that my opponent would heavily fortify it. Then I was able to increase  the troops at all the other castles."
5,6,7,8,10,14,15,15,15,0,meh
5,6,7,8,9,11,12,13,14,15,This should perform well on average but is far from optimal.
5,6,7,8,9,11,12,13,14,15,"It seemed to me that the optimal strategy would be to take advantage of each weakness in my opponent's line, so something like 10 at each castle makes some sense, but I also wanted to weight the more valuable castles more heavily, so I started with a base of 5, and distributed the remaining 50 across the rest, weighted by castle value."
5,6,7,8,9,11,12,13,14,15,"It's a bit of a weighted average.  I first deployed 1-10 soldiers based on point values, one to Castle 1, two to Castle 2, et cetera.  That left 45 soldiers.  I then distributed the rest evenly, sending 4 more soldiers to each castle, and then sending the last 5 to the top 5 castles.  I figure some adversaries will be more top-heavy, and this way I might win some of the middle and lower castles and make it a close contest.  This distribution would also beat those who went for a pure average 10-man-per-castle deployment."
5,6,7,8,9,11,12,13,14,15,this will defeat those who use an even distribution and those who give slight preference to higher value castles.
5,6,7,8,9,11,12,13,14,15,"By giving more clout to the larger castles, while still packing a punch with the castles of fewer points, this set up guarantees at least something to walk away with."
5,6,6,15,15,25,25,1,1,1,"thinking im gonna do something like this: 1 on 10 1 on 9 1 on 8... and distrubte 97 more from 7-1...if you can win 7-1 (28 total) over 10-8 (27) total, you are in good shape. so ignoring top 3 and focusing on lower 7-1 is a better strat. if others feel the same as me, it will feel be sweet to perhaps steal one of the ""top 3"" with only one soldier each. "
5,6,4,5,24,20,20,10,3,3,cuz I'm a legend boi go to the outsider website
5,5,15,15,15,15,15,15,0,0,"I don't have to beat everyone, just a lot of people. I figure most people will put a lot of emphasis on castles 9 and 10, so I'm just ignoring those two, in hopes of winning all of 1-7 or 8 and most of 1-7. I was originally gonna distribute evenly between castles, but thought that might be common so I decided to put slightly more than an even distribution in most castles, and leave fewer in 1 and 2, which I could lose if I win 8."
5,5,11,17,4,7,25,10,11,5,Random
5,5,10,20,20,20,20,0,0,0,
5,5,10,10,20,25,25,0,0,0,I assume most would put their soldiers at the higher point castles. I placed mine at the top 7 which would give me the most points.  
5,5,10,10,15,15,20,20,0,0,"I figured a lot of people would load up on 9-10 so I focused on lower value, but easier to win (I hope) castles"
5,5,5,10,15,25,35,0,0,0,"If my opponent goes for high value, I could possibly win all the low values, which are collectively more points."
5,5,5,5,20,30,30,0,0,0,"I win 28-27 if I win castles 1-7, figuring most people will go for 8, 9, 10"
5,5,5,5,15,15,15,25,5,5,Al achunte
5,5,5,5,11,12,13,14,15,15,lose the first five but get the rest
5,5,5,5,10,20,2,2,30,16,"I seeded the competition with some sensible strategies, then added other strategies that tended to do well against those, then added others and so on. "
5,5,5,5,10,10,15,25,19,1,"if I can win 1-7, I win. Figured many people will try to spend a lot of resources on 8,9 and 10."
5,5,5,5,10,10,10,15,15,20,Intuitive
5,5,5,5,5,18,25,1,30,1,"To get the 28 points needed to win, you need to win at least 4 castles. I figured many people would devise a strategy to focus on winning 4 or 5 mid-to-high valued castles (meaning they'd split their troops into 4-5 main groups of roughly 20-25 troops) with maybe a few left for smaller valued castles. I, instead, decided to focus on 3 main targets (9, 7, and 6) and try to make up the difference by winning several of the lower-valued castles by sending 5 troops each to castles 1-5 (assuming most people will be willing to sacrifice several of them)."
5,5,5,5,5,8,15,30,15,7,"not very good at game theory, but I wanted something where I could leave enough to clean up on some low value targets if people ended up being too concentrated, focus on some powerful hitters, and not waste too much on folks who wanted to hold hard to the top castle, but catch anyone who does a complete pump fake on ten. Let's see if it works!"
5,5,5,5,5,5,30,30,5,5,Fortune favors the brave!
5,5,5,5,5,5,30,30,5,5,Fortune favors the brave!
5,5,5,5,5,5,27,30,6,7,Updated. Fortune favors the brave!
5,5,5,5,5,5,5,5,5,55,Played 20 test rounds with some friends and this one won the most frequently. Strong chance of winning 10 and at least 18 leftover points (17.5 for the tie). Prevents the 1 and 2 strategy on not desirable numbers. 
5,5,0,0,0,0,0,30,30,30,You need a minimum of 4 castles.  Want to try to ensure the top three and gives a good shot at lower.
5,2,1,9,4,10,6,20,21,22,"I took 10,000 random troop deployments, and found the winning deployment.  I did that 10,000 times, to generate 10,000 good deployments.  Then I generated a few million random deployments to test against those 10,000 good deplolyments -- and this one won all 10,000!   I cannot understand why this worked, but I'll go for it."
5,1,2,3,4,10,6,7,8,9,
5,1,1,1,1,1,1,26,30,33,"OK, I figured that I need 28 points to win.  Thus, taking castles 10, 9, 8, and 1 would suffice.  First, I allot one soldier to each castle, should the enemy king omit any.  With the remaining 90, I allotted them according to the proportional value of my target castles relative to the required 28 points, calculating this as, approximately 33 additional soldiers to C10, 30 to C9, 26 to C8, and 5 soldiers to C1.  Having sent off my army, I prepare a huge victory parade that will have the largest crowds ever, no matter what the park service says."
5,1,1,1,1,1,1,26,30,33,"There are 55 total victory points in the game, therefore a player needs to get 28 points.  I assume that my opponent will then choose a strategy that only sends troops to castles to achieve the minimum of 28 points and not send any troops to the other 27 points.  Regardless of which strategy he chooses my strategy will beat any strategy that chooses to completely ignore castles."
5,1,1,1,1,1,1,25,29,35,"In order to win the war, I need to get more victory points than my opponent.  With 55 total victory points at stake, I need 28 victory points to win.  The top three castles are collectively worth 27 (8+9+10) points, so if I win those, I only need to win one more castle to win the war.  The vast majority of my soldiers go to castles 8, 9, and 10 since they are the most valuable.  I send five troops to castle 1 because I doubt most of my opponents will send many troops to the least valuable castle.  I send one soldier to each of the remaining castles (2-7) just in case my opponent neglects to send any troops there.  These six soldiers don't hurt me much in other areas.  Overall, I think this strategy is the best way to win the race to 28 points.  "
5,1,0,11,14,13,14,11,17,14,"Simulation! I created 10,000 possible deployment strategies. For each one, I generated a scalar, u, from Uniform(0.1, 10). Then, generate a set of probabilities, theta, from Dirichlet((sqrt(u), sqrt(u), ..., sqrt(u))). Finally, generate the troop deployment from Multinomial(100, theta). Then, I paired off all possible deployment strategies and found the one that won the most. The top few seemed pretty similar, but might as well go with the best of the best, even if it's probable that a slight change will beat this particular one."
5,0,11,14,16,0,22,32,0,0,Put all soldiers on acombination that adds to 28 and hope the opponent chooses a more even distribution
5,0,10,0,0,0,20,30,35,0,Hope to win majority of points available and no more.
5,0,0,0,14,21,25,1,32,2,Focus on five castles to reach 28 points.  Try to disrupt strategies going after low or high castle numbers.  Capture uncontested high-value castles. 
5,0,0,0,0,0,0,27,31,37,"The three most-valuable castles are worth 27 total, and the 7 least-valuable are worth 28.  So making a strong claim to 27 points and a weak claim to the 28th point seems like a good distribution.  The vulnerabilities can be exposed, though, by a distribution that weights castles 2-7 as moderately important, and emphasizes a  strong attack on one castle in the 8-10 range.  I just have to count on my 8-10 range being fortified enough and few enough other people being crazy enough to send 5 soldiers to a castle worth 1 point."
5,0,0,0,0,0,0,25,30,40,Only need 28 total pts to win the battle
4,9,9,13,15,15,17,18,0,0,Concede the top 2 to overload the rest.
4,8,11,14,17,21,25,0,0,0,Needed to get 28 total points to win.  Used win probability (number of soldiers allocated vs. the expected distribution) to allocate soldiers in the way that had the highest probability of getting 28 points.
4,8,11,14,17,20,23,1,1,1,"Castles 8, 9, and 10 are likely to receive heavy troop deployments because they are worth the most points. 1 + ... + 10 = 55. You can afford to completely ignore Castles 8, 9 and 10 (Sum: 27) as long as you can guarantee a high probability of winning Castles 1 through 7 (Sum: 28). I determined that having a deployment of better than 3 soldiers per point for Castles 1 through 7 (generally 3p + 2, with the exception of Castle 1) would give me a very good chance of winning them all.

I also wanted to cover my bases a little by sending 1 troop to Castles 8, 9 and 10 in case there was an implicit requirement to deploy a troop to every castle, and also to catch out the other people would would be doing something similar (ie - ignoring the high-end castles to focus on the low-end castles)."
4,8,10,10,15,20,27,2,2,2,Giving up 27 points to hopefully win the remaining 28. I stuck two troops in the big three in case anyone else had a similar idea.
4,7,11,14,18,21,25,0,0,0,I considered having each soldier is fighting for (roughly as it's rounded) 1% of the total points (e.g. 2-4-5-7-9-11-13-15-16-18) but that seemed to be dominated by more concentrated strategies that ignore some castles. So I doubled up on the lower end to try to win all of 1-7 and ignored the top end where I'm hoping most people will put their soldiers
4,7,11,14,18,21,25,0,0,0,Castles 1-7 are worth more than 50% of the points.
4,7,11,14,18,21,25,0,0,0,"Concentrate on lower values, expecting others to overplay on higher values. If I win these, I get just enough to win, no matter what happens at the higher value castles."
4,7,11,14,18,21,25,0,0,0,"I yield the 3 juiciest castles to my opponent, and try to win all 7 remaining castles. I deploy my soldiers to each castle in proportion to its prize (rounding to the nearest integer)."
4,7,11,14,18,21,25,0,0,0,"Get there firstest, with the mostest! Race to 28 pts."
4,7,11,14,18,21,25,0,0,0,"There are 55 available points, the winner needs 28.  Working my way up, I allocated my troops according to the relative value of each castle (1/28 = 3.57% = 4 troops to Castle 1; 2/28 = 7.18% = 7 troops to Castle 2; etc.) until I ran out."
4,7,11,14,18,21,25,0,0,0,"Placed a higher than normal interest in the Castles 1-7 and allowed enemy to capture the three highest valued castles. By capturing the weaker castles, I can earn 28 points, while only surrendering 27 points to the enemy. This is under the prediction that my opponent will target the higher valued castles, which many not be the case. I'm an irrational S.O.B."
4,7,11,14,18,21,25,0,0,0,Need 28 points to win. 100/28 = 3.5. People are more likely to over allocate to the top value castles. If I win 1-7 victory castles that makes 28 points. Allocate amongst those by the weighted value of each one.
4,7,11,14,18,21,25,0,0,0,"I can win if I take all castles 1-7, so I allocate all troops to those castles proportional to their value... and hope other people allocate to castles proportionately, but allocate at least some to 8-10."
4,7,11,14,18,21,25,0,0,0,To win castles 1-7 (total Victory points 28)
4,7,11,14,17,21,26,0,0,0,Avoid being beaten by concentrated bets. This strategy is fragile in that it seeks to win with minimum needed (28) while spreading across the most selections to do so.   It chooses not to defend the three most valuable castles in an attempt to win all others. Anyone taking a variation of this approach will likely lose close decisive battles for the ultimate margin of victory against me - I hope. 
4,7,11,14,17,20,24,1,1,1,"There are 55 points in total, so you need 28 to win. The lower 7 castles offer enough points to win, and it made sense to me to provision them with forces in a manner roughly proportional to their value. The remaining three castles I gave 1 to... because it seemed like a good idea at the time."
4,7,10,15,17,22,25,0,0,0,"realizing I needed 28 points to win, (55/2 rounded up). I originally was going to put my troops on 4 key castles (9,8,7,4). needing 28 points i realized that i had 3.57 troops per point. I was worried though that was putting too much into on basket, so i spread out over the bottom rung (1 through 7) otherwise known as the lowball strategy. "
4,7,10,14,17,21,24,1,1,1,"55 points total. I assume submissions are more likely to load troops on castles 10,9,8, and If I win all castles 1-7, that is sufficient. I allocate 1 to each 10,9,8 so the opponent does not get a free win, and the rest are proportional the that castle's value over the remaining total point value of castles 1-7. (= Value x (100-3) / 28) "
4,7,10,14,17,21,24,1,1,1,"Trying to win just enough castles, but also foil plans that use zero on some large castles."
4,7,10,13,16,19,22,2,3,4,"With 55 possible points, an army needs a total of 28 points to win. Thus, I can sacrifice 27 points. The quickest pathway to do so is to allow my opponent to take the castles worth 10, 9 and 8 points, for a total of 27 points. Therefore, I need to take all other castles in order to win. Thus, I split my troops along the other castles with equal force, assuming that my troops on Castle 1 is 7 times more valuable than my troops on Castle 7, 6 times more powerful than Castle 6, so on and so forth, as I will need 7 times as many troops to dedicate to Castle 7, as it is 7 times more powerful. It was then easy to distribute my troops along those lines, leaving 16 troops. After adding 1 troop to each Castle as extra defense, I added 4 to Castle 10, 3 to Castle 9, and 2 to Castle 8 in order to slightly defend against my own strategy. Knowing I had 9 extra troops, I applied the same logic to my three extra Castles, saying that Each Castle 8 soldier would be 10/8ths as strong as Castle 10. Then, I simply rounded in order to get my troop allocation to be as close as possible to those fractions. "
4,7,9,12,14,16,18,20,0,0,1-8 majority of points
4,6,8,12,16,21,33,0,0,0,The winner needs 28 points so I focused all my resources on towers tha will get me 28 points while avoiding the largest castles that most people would focus on.
4,6,8,10,12,14,17,19,4,5,"Concede the most valuable, try to pick up most of the rest."
4,6,7,9,11,13,15,17,18,0,"A basic strategy could be to deploy an average number of troops to each castle weighted by point value, in which case one would deploy 1.82 troops per point the castle was worth. Given that Castle 10 is the highest profile target, I expect my opponent to commit an above average number of troops to it. I submitted 0 troops to Castle 10 so they would waste any troops they committed to 10 over the average. Instead, I committed an above average number of troops to every other castle, distributing the average number of Castle 10 troops (18) over all the other castles (2 per castle)."
4,5,6,13,13,13,13,13,18,2,"Abandoned first castle since it would probably face strong opposition that could be better distributed elsewhere, then put one guy back to catch anyone who did what I thought to do, Then tried to put roughly equal soldiers at the rest since I'd need them all (besides the last couple that are worth very little)"
4,5,6,8,11,24,42,0,0,0,Trying to win 28-27
4,5,6,7,8,18,22,26,2,2,Thought most people would go after high value targets and I prefer a quantity over quality approach. Sent a few troops at top castles to counter a similar technique.
4,5,6,7,8,10,13,19,28,0,"By trying to determine the optimal strategy, then countering that strategy. And likely failing"
4,5,2,19,18,19,3,4,3,23,"I tested randomly generated deployments against others, and the winners advanced to the next round vs. a newly generated random.  the one with the most wins is the submission.  I tried to test the winner against all possible deployments, but O^2... it ran all weekend and still hasn't finished haha."
4,4,19,4,19,4,19,4,19,4,"I decided to focus only the odd castles, except for Castle 1. Focusing on four castles and leaving a minimum of 4 in every other castle, that leaves 19 in Castles 3,5,7,9."
4,4,6,3,6,9,25,9,17,17,I ran a competition among me and my friends.  Each person submitted 3 solutions.  I wrote a python program to pit everyone's submissions against everyone else's.  The person with the best board actually created his by creating over a 100 sample boards of his own to pit his trial boards against.  This is a modification of his winning submission for our competition.
4,4,4,8,13,17,20,30,0,0,Try to get the minimum 28 to win by focusing on the lower ones.
4,4,4,6,8,13,15,22,24,0,"I thought about different possible strategies, then devised a plan that beat most of those strategies. "
4,4,4,4,12,20,25,25,1,1,"There are 55 total points, so you need to get 28 to win.  You probably want to send somebody to every castle in case your opponent skips it, and that does not use up many troops anyway.  After that, my goal was to optimize getting to 28 without losing a ton of troops fighting over 9 and 10.  If I win 5,6,7,8, that gives me 26, and so I sent a few extra troops to both 2 and 3 so that I can win one of those as well.  I also sent a few extra troops to 1 and 4 in case somebody plays heavy on 8, 9,and 10 and sends an extra few troops downstream to pick up a point."
4,4,4,4,4,24,4,26,26,0,Dunno really
4,4,4,4,4,4,34,34,4,4,No specific reason.
4,2,2,15,3,23,4,4,5,38,"To maximize victories, the obvious plan is to ""win by a little, lose by a lot"". I suspect many will do this by concentrating soldiers in castles totaling just over 27 (half), and deploying very few at others. Therefore, I counter by deploying *more than should be enough* at castles totaling 21 points, virtually assuring victory at those, but distributing enough to win at least some of the castles where they may often be 0 to 4 armies. "
4,1,6,1,11,5,16,10,30,16,Picked something resembling a pattern and hoped.
4,1,1,16,21,25,29,1,1,1,
4,1,1,1,1,1,1,30,30,30,"8-10 are 27 points and 1-7 are 28 points. This is to attempt to ensure a win at 8,9,10 and 1. If you bet me at either 8,9 or 10. I will win the bottom 6 castles and win. See if it works."
4,1,1,1,1,1,1,30,30,30,Aim to win the big three and then the smallest one to get exactly half the points plus 1.
4,1,1,1,1,1,1,28,30,32,"Need 28/55 to win, 10+9+8+1 is 28, throw a few in the others in case they realize that and don't put any in 2 through 7."
4,1,1,1,1,1,1,26,30,34,"You need 28 victory points to ensure victory over your opponent. Figured the most efficient way to do this was to hit that number contesting as few castles as possible, so concentrated on the highest value 3 castles and then only needed one more to get 28 so put a few extra in the 1. 

I hedged slightly by putting one soldier in each of the other 6 castles in case my opponent chose to not send anyone there. 

I distributed the remaining troops based on % of needed victory points per castle. "
4,0,12,0,19,0,30,0,35,0,"Simulation, using mode castle placement, where probability of assigning a soldier to a castle is based on points but ignoring even-numbered castles."
4,0,1,1,5,10,18,19,22,20,"I wrote a short code in Matlab to create 100,000 random strategies to see which won the most. To simplify the code, I allowed the number of troops to be continuous, and then I just rounded off the values of the best one."
4,0,0,0,1,1,1,30,31,32,"Heavy value on 10,9,8, and 1 as you would only have to win those 4 castles to win any battle."
4,0,0,0,0,0,0,32,32,32,"Since I needed to win just over 50% of the possible 55 points I put all my men into the 4 castles that would earn 28 points and conceded the rest to my enemies. I figured this would allow me to concentrate my forces on castles that would guarantee me a victory if I was able to capture them. I know this is a risky (foolish?) strategy because I'm giving my enemies 27 points and failure to capture my 4 target castles would guarantee defeat. I'll be interested to see how my gamble/this game plays out. 

""Once more unto the breach"""
4,0,0,0,0,0,0,32,32,32,"All in, just like in Poker - I bet you can tell I lose a lot of money :("
4,0,0,0,0,0,0,32,32,32,"I decided to go all in on a single strategy instead of hedging. You need to conquer a minimum on 4 four castles to win. I am putting all my soldiers into those four castles, so I want at least one of them to be uncontested to free up soldiers for other castles. There is only one such group of four that includes the least contested castle. That is (1, 8, 9, 10). I put the minimum force towards 1 that I thought could gain me victory relatively often."
4,0,0,0,0,0,0,29,32,35,
4,0,0,0,0,0,0,28,32,36,"I want to maximize my victory points, that is, with the least number of soldiers.  The higher the castle, the more troops needed to secure a victory point.  To win, I need more than half of the total victory points, which is 55 (to win, I need 28).  To  achieve this, I selected the fewest castles that will allow me to get 28 victory points, that is: castles 10, 9, 8 and 1 (10+9+8+1=28).  So I need to distribute 100 soldiers in these 4 castles and let opponent take all other castles.  I weighted the victory points to win vs the amount of soldiers, ie castle 10= 10/28*100=35.7, or 36 , castle 9= 9/28*100=32.1 or 32, castle 8= 8/28*100=28.57, or 29-1. and castle 1 is 1/28*100= 3.57 =4.  I assumed castle 1 would be uncontested, but ensured at least its value of 4."
4,0,0,0,0,0,0,28,32,36,"To choose a strategy, I have thought about two axes. The first one is the tendency of players to use balanced strategies such as x y x y??_ with x+y =20 as a response to the simplest strategy of 10 10 10 ??_  the second one is to concentrate troops on castles with 28 as value sum (the number needed to win a battle). Each castle i with a number of troops proportional to its value i.e. 100i/28. Using these two hypotheses, castles with highest values would have fewer troops than their values required (using the first axe). The best strategy would be then (using the second axe) to use a set a castle with sum values 28 and with the highest values possible. We end up then with using Castle 10,9,8 and 1 (sum is 28). Proportional allocation would be 36,32,28 and 4 (sum is 100)."
3,11,11,11,11,21,26,2,2,2,"I need to win 28 points to win a war, If someone evenly distributes I beat them. Most other scenarios I win. "
3,9,14,14,10,3,16,14,16,1,"I wanted a quick programming challenge so I put together a small python notebook to create 200,000 different deployments, ran all the match ups, and spit out a winning deployment. I ran the program five times to get five different deployments, matched them all up against each other and picked the one that had the highest average number of points. Not sexy, but it was fun."
3,7,11,14,18,21,26,0,0,0,Point weighted distribution of troops for the lowest 7 ranked castles (which constitute the majority of the points in the game).
3,7,11,14,0,0,0,29,0,36,"I've no real knowledge of game theory so I'd imagine mine is extremely primitive but it was based on the idea of attempting to win exactly enough points to have a majority and not contest the other towers. Obviously there are a variety of combinations that come to the 28 points needed. I then calculated how many troops should go to each tower proportionally based on the value of the tower relative to the target value of 28. As for which of the many combinations adding up to 28 I selected? Well I stook my finger in the air and picked (10, 8, 4, 3, 2, 1), as I felt it had a nice balance of covering the Highest value tower, but also covering a decent spread of other towers."
3,7,10,14,18,21,24,1,1,1,Taking the bottom 7 castles gets you to 28 which the minimum required to win
3,7,10,14,17,21,25,1,1,1,"Modified probability - allocated 1 soldier to each of the highest 3 points, to take advantage of anyone who was leaving those castles undefended.  Other than that, allocated based upon win probability."
3,7,10,14,17,21,25,1,1,1,"I went through several iterations, then my brain started hurting trying to anticipate what other players would do. Here the (lame) logic of my deployment: Castles 1-7 are worth 28 points, enough to win. So I proportionally allocate all of my troops over those castles. Except, I send a single troop to each of 8, 9, and 10, in case my opponent fails to send any troops there and I can pick up an easy win. I'm not happy with it, but oh well."
3,7,10,14,17,21,25,1,1,1,I want a shot at each castle in case someone sends 0.  Beyond that the bottom 7 castles give a high enough score to win and I proportionately distributed the soldiers by point value.
3,7,10,14,17,21,25,1,1,1,"The goal is to get at least 28 points so I used a simple equation for how many soldiers to put in each castle: 100??x/28 where is x is vp of each castle. I then rounded all of the decimals down and put those number of soldiers in each specific castle. I spent the last three soldiers in the 8,9,and 10 castles to win if there were no soldiers from the other side. There are other combinations of castles which get you 28 points but I chose to compete for castles 1-7 because I thought castles 8-10 would be more competitive so I avoided them."
3,7,10,14,17,21,24,1,1,2,"The simplest strategy is to distribute troops evenly, in proportion to the 55 points available.  That yields the average expected value of ~ 1/2 point per troop.

But, we only need to get to 28 to win.  I played with a number of strategies where people won only high value castles (25 points on each of castles 1+8+9+10 wins, for instance).  But, those strategies leave vulnerable the other castles for picking by a sole attacker.

So, I chose to attack every castle with at least one, but to focus on the lowest seven for a win (1+2+3+4+5+6+7 = 28).  I threw one person on castles 8-10, so if they are left undefended I have a chance of picking off a high value one.  But I have 97 points distributed on the lowest value seven.

It can be beat, but not by the first several options in game theory.

Like the quick question about 2/3rds of the average guess, there is no theoretical right answer.  We have to assess how many people will pick which strategy, and then pick a strategy in response to that."
3,7,0,14,0,21,26,29,0,0,"The strategy in blotto games is always an attempt to win each castle that you win by as few soldiers as possible, while losing the castles you lose by as many as possible, in such a way as to get more than half of the available points (here the target is 28 points).  I decided to chose the castles adding up to 28 points that I thought the fewest people would put significant resources in to securing, and roughly allocate my 100 armies to those castles proportionally to their point values, giving up completely on the other castles."
3,6,10,14,18,22,27,0,0,0,I focused on winning exactly 28 points against as many likely strategies as possible.
3,6,10,14,18,21,25,1,1,1,"Aim to win 1-7, conceding 8+"
3,6,10,12,16,21,26,2,2,2,"The idea is that other warlords won't emphasize the low value castles as much as much as I do, and if they do they won't hedge as much on the high value ones. "
3,6,9,12,15,18,21,16,0,0,"There are a total of 55 victory points on offer, so the aim is to win at least 28 points as often as possible. I chose to aim more heavily for castles 1-7, as I thought they are likely to be less often well protected. "
3,6,9,12,15,18,21,5,5,6,"First idea was that troops should be deployed in line with castle value.  Second idea was that people will overweight top castles so could easily burn troops there.  Third idea was that winning first 7 castles wins 28-27.

Keeping first principle can put 3 troops per point on first 7.  Not enough for one troop per point after this.  Distribute remaining 16 over last 3 to win over people who write off 8, 9 or 10, so 5 each and 1 left over.  Stick this extra one on 10 as that is the highest value castle (not very consistent with earlier but decision where to put last one required)."
3,6,9,12,15,17,20,16,1,1,"You need to achieve 28 points.  Most people will try for 10+9+8+1, but I will try for 1+2+3+4+5+6+7.  If they steal my 1, I will steal their 8."
3,6,9,12,14,16,18,22,0,0,"I call this the Crowe-Nash deployment strategy, in honor of the nonsensical description of game theory in A Beautiful Mind:  https://www.youtube.com/watch?v=LJS7Igvk6ZM .
The idea is that castles 9 and 10 are the blonde girl and everyone will deploy troops heavily to get them. By ignoring them completely, I make a heavy play for the other castles that add up to 36 of the 55 available points."
3,6,9,11,14,16,18,20,1,2,I thought I would need around 34 points to win. Giving up the highest 2 castles means that I have around 36 points and I put around 3 times the number of points in the remaining castles. I then put some in castles 9 and 10 just in case a enough people try giving them up.
3,6,9,3,3,18,3,24,27,4,"Making an evenly balanced effort to win the minimum 28 points needed to win. 84 soldiers (3 per point) committed across 5 target castles. 

The remaining 16 are then spread across the remaining castles to pick up easy points from more aggressive strategies which leave some castles unchallenged or send only 1 or 2 soldiers.

If an opponent commits significantly >3 soldiers per point they are targeting, then they aren't targeting enough to win.

If they commit <3 per point they are targeting, I will win out on my targets.

The greatest threats I face are similar strategies, but slightly more aggressive. However, these strategies are likely to suffer more against slightly more random / less calculated strategies, which I am hoping are common enough to protect me.

I also deliberately avoided targeting 10 and 5 as the more salient target castles."
3,6,7,9,11,13,15,17,18,1,"Started with allocating troops according to each castle's fraction of the total points available. Assuming that other smart readers would do the same, and that some readers would focus on castle 10, decided to mostly ignore castle 10. Added 2 to each castle other than 10, then took 1 back from castle 1 and gave it to 10 so it wouldn't be completely uncontested."
3,6,7,9,11,2,27,31,2,2,"I suspect that many strategies will have castles with either 0 or 1 armies and will focus their armies on castles worth about 28 points. Other strategies will distribute their armies more evenly. 

If one distributed the 100 armies evenly by the points of the castle, one would get the following: 2, 4, 5, 7, 9, 11, 13, 15, 16, 18. If one distributed the 100 armies according to castle points but focused on just castles worth 28 points, the armies would be some combination zero values and numbers like: 4, 7, 11, 15, 18, 22, 25, 29, 33, 36 (with slight variations due to rounding). 

In general, I will win over a given opponent if I tend to win castles by having only slightly more armies than my opponent, but lose castles by having much less than my opponent. Each of my 10 castles will have armies designed to slightly beat one of the following opposing strategies for that castle: an essentially undefended castle (castles 6, 9 & 10), an equally distributed castle (castles (1, 2, 3, 4 & 5), or a focused attack castle (castles 7 & 8). This gives the following values: 3, 6, 7, 9, 11, 2, 27, 31, 2, 2.
"
3,6,6,15,2,5,22,37,2,2,Figured most people would load up 9 and 10. Minimum deployment of 2 to some castles to account for the people who would presumably deploy 0 or 1 to them.
3,6,6,11,11,16,21,26,0,0,"I think most people will stack troops at 9 and 10, so I will try to give those up and win almost all of the other castles"
3,6,1,13,1,21,25,28,1,1,"I decided to put the vast majority of my resources towards securing the 6 castles whose total point value is 28, adequate to win, that I thought fewer other people would be pursuing, while only allocating 1 each to the remaining 4 castles, which I thought others might be trying hard for."
3,5,8,13,21,25,25,0,0,0,"Going for middle ranked castles, and hoping to pick off lower castles for few troops"
3,5,8,13,18,24,29,0,0,0,"My approach is not to win as many points as possible, but to take the absolute minimum points to win the majority of the time. In this game you are forced to make assumptions about the enemy. I'm hoping that castles 8, 9 and 10 will be ""lucrative traps."" With the higher concentration of points, the enemy goes at them with a lot of resources. Instead of trying to take those with my own resources, I will sacrifice those 3 castles to try to pick up the remaining 28 points from the rest of the castles. Essentially, the idea is that the enemy will not have enough resources left for the smaller 7, and I can (hopefully) win them all with small margins."
3,5,8,11,14,16,21,22,0,0,"I am counting on most people to invest heavily in trying to win castles 9 and 10.  Therefore, I am ceding those castles to them in hopes that my additional troops can win the remaining castles.  This gives me a very narrow margin of victory (I can only lose 8 points in the remaining 8 castles), but against a top-heavy attack, I like my chances.  I'm not convinced that this is the best strategy, but I know that in large-scale competitions, it is good to be an outlier (think of the NCAA tournament:  pick the favorites, and you are likely to finish in the top 10%, but very unlikely to win the whole competition).  "
3,5,8,10,13,1,26,30,2,2,
3,5,7,13,16,19,17,10,7,3,Trying to buy low
3,5,7,11,14,17,20,23,0,0,I predict people will over-deploy on the biggest castles. Thus I can save these troops and use them to win all the other castles. I added a bonus troop to castles 7 and 8 in case someone thinks like I do.
3,5,7,9,11,13,15,17,20,0,Concede castle 10 in hopes that opponents will deploy large troops there.
3,5,7,9,11,13,15,17,19,1,"Punt on Castle 10, try to take the rest."
3,5,7,9,11,13,15,17,19,1,"I thought about splitting my troops roughly equally according to the castles' value. Then I decided to sacrifice Castle 10, which I expect most people to fight over, and get a bit of a boost everywhere else. But I left 1 soldier at Castle 10 in case some people send none there. 

By the way it would be cool if you published not just the winner but the entire ranking table so we could all see how we did."
3,5,7,9,11,13,15,17,19,1,x 2 +1 
3,5,7,9,11,13,15,17,19,1,"Intuition as a game designer - trying to stay 2 steps ahead. Also, liked the pattern."
3,5,7,9,11,13,15,17,19,1,
3,5,6,10,20,23,30,1,1,1,"There are 55 pts in the game -- if I have 28, the game is done (and I the winner). Thus I hope to win castles 1???7 ... that results in 28 pts. I abandon castles 8, 9, 10 ??? the sum of three totaling 27 pts ??? assuming most players will seek the big numbers first. I do send one (unfortunate) solo man in the case the castle is indeed empty. If so, free pts for me! It is my hope I win out across the bottom 7 castles. Little room for error, but such is the case for most wars."
3,4,11,13,16,21,27,2,2,1,Scoring 1 to 7
3,4,9,10,14,6,11,6,9,18,I generated random troop distributions in numpy and ran tournaments with 25 distributions each. I did tiers of tournaments where the winners of 25 tournaments with random distributions were put in another round of tournaments and so on 4 times.
3,4,8,0,20,0,30,35,0,0,to not deploy forces to the most valuable castles where there would likely be the most competition.  Place strength on mid value and low value targets to reach goal of 26
3,4,5,18,22,20,21,2,3,2,
3,4,5,6,7,8,22,22,22,1,"I expect many people to try hard for Castle 10, so no point in gunning for that.  I put one there just in case some opponents abandon it entirely.  7, 8, and 9 together will get me most of the way to victory, so I put the bulk of my troops there, and then the remainder on the rest of the castles in diminishing order, to try to pick up the few extra points I would need."
3,4,4,5,6,7,9,11,17,34,Simple weighting based on 1/#of points available.
3,4,4,5,6,0,0,24,26,28,"Started with 10 on each castle (assuming average of random assignment), and then made a ""k-1"" deployment that would beat that deployment, and then a ""k-2"" that would beat the ""k-1"" deployment, etc., out until k-10, then made some minor adjustments."
3,4,4,3,6,11,20,25,11,13,Beat people who oversubscribe to 9 and 10 by stacking 7 and 8 and a reasonable distribution for other castles.  But guard against others giving up on 9 and 10 by still putting a chunk there.
3,4,0,0,1,0,2,0,7,83,"I ran a few simulations in MatLab, starting with warlords who randomly assigned their soldiers and then using the most successful warlords of each previous generation to bias the assignments of the next. This is a rough average of some of the winning strategies after a few hundred rounds."
3,3,19,3,3,3,3,21,21,21,"You get more points for winning the top castles than for winning the lesser castles.  There are a total of 55 points available.  Winning the top three and a fourth castle gets you the win.  Because it takes four winning castles, my hunch is that a lot will go 25 to 4 castles.  Against those individuals, I am conceding the top three and leaving enough elsewhere to pick up the bottom 7.  The potential weakness is somebody who goes stronger than 25 to the top three and hope to pick up a fourth (there is a one in seven shot that I will have picked the right number to go strong against).  A strategy of wining the bottom 7 (and conceding the top three) would involve 14 to each castle.  By having more than 15 to one of the bottom 7, I hope to pick up the extra."
3,3,13,13,3,13,13,18,3,18,
3,3,10,12,12,15,17,22,3,3,"Don't want extremes, but want enough that if people put 0 i get something. Want high to low numbers for 8-2"
3,3,7,3,3,3,22,24,3,29,hoping that many people put in 1 or 2 troops on the minor castles.  
3,3,6,7,3,15,10,30,20,3,
3,3,5,8,8,11,11,14,17,20,"Concentrating on a 4 castle = 27.5 points strategy.  There are 9 possible combinations of 4 castle win strategies.  Castle 10 appears 7 times in those 9 combinations.  Castle 9 appears 6 times... etc down to castle 1 which appears once.  I weighted the number of troops per castle by the relative number of times a castle appears on the four castle win strategy.  

9 combinations:
10-9-8-1  10-9-7-2  10-9-6-3 10-9-5-4  10-8-7-3  10-8-6-4  10-7-6-5  9-8-7-4  9-8-6-5  

normalizes to roughly 2.7 troops per castle per combination "
3,3,5,5,7,7,30,30,5,5,"Conquer the low castles, spam a few higher ones."
3,3,4,6,8,11,15,21,19,10,"I would like to win 7,8,9 and least get a draw in the small ones"
3,3,4,4,4,4,4,35,4,35, 
3,3,3,19,3,24,3,3,3,36,"focus on 3 castles, disrupt other strategies, capture uncontested castles"
3,3,3,15,15,15,20,20,3,3,My cat chose for me 
3,3,3,13,25,26,27,0,0,0,Ran through a couple of scenarios in a simple model I built in Excel. I liked this one because it leveraged the strategy of keeping the enemy from winning vs the strategy of trying to win.
3,3,3,8,9,20,22,15,12,5,"Because it was required to enter the contest! My decisions were based on the following train of thought:
I decided nothing was worth less than 3, because there are likely a number of people who will do a minimum, I thought i could pick off a number of castles with a slightly higher than expected valued minimum. Of the Remaining 70 soldiers, I randomly assigned 10 of them for three tower bands from 4-6 through 8-10, and then with the twenty remaining soldiers i divided them randomly in groups of ten for the band from 5-8 and 4-7. 
This all SOUNDS very academic, it is not. It is random. And yet...i spent like an hour deciding if it was random AND correct. Because damn you, damn you and your making me procrastinate Oliver!! YOU WILL RUE THE DAY! "
3,3,3,4,8,8,13,16,19,23,weighted points and submitted more troops to most valuable but consistently gave troops to even least valuable. 
3,3,3,3,11,16,21,34,3,3,"I think a lot of people will want to over-commit to large castles.  Some may choose to send everything they have at the top four or five and ignore the rest.  Some who think another step further may send one to every castle to win against those who send zero -- opportunity cost of sending one troop isn't that high. So, my strategy is to send 3 to every castle hoping to win all those simple and low-contested battles.  Then, I'll mostly ignore castles 9 and 10, assuming that's where most people will focus large amounts of troops, and allocate the rest of my troops in a roughly logrithmic fall-off starting with castle 8.  Under the thought that most people will pick numbers that are psychologically pleasing (33 (as one third) or ending in a 5 or a 0), I allocated the rest of my troops starting with 34 troops at castle 8 and falling off from there."
3,3,3,3,7,16,18,21,23,3,"It's worth it to put in at least 1 troop to contest every castle; I put in three to beat the 1 and 2 choosers. Ignore 10, most people will be top-heavy, so it's not worth trying to win there. "
3,3,3,3,7,12,22,22,22,3,"focus on winning 7-9, ignore 1-4 and 10 (unless other person ignores, then win them); numbers are 2 above round numbers (1, 5, 10, 20) to beat people who would one-up those"
3,3,3,3,3,38,3,3,3,38,"focus on 3 castles, disrupt other strategies, capture uncontested castles"
3,3,3,3,3,21,31,31,1,1,"I suspect many people will place great weights on Castles 9 and 10, so I prioritized the next few highest castles. Further, I anticipate many people only awarding one troop to the smaller castles, and some placing two troops to counter that strategy, so I reasoned that placing three troops at Castles 1-5 would let me win some portion of those castles."
3,3,3,3,3,16,20,22,24,3,"I went with a ""barbell"" approach. I invested at least 3 soldiers in every castle because I figure I'll be able to win some castles with low investments. And then I picked four castles for larger investments. I thought people would invest lots of soldiers in castle 10, so I did not invest a lot of soldiers there."
3,3,3,3,3,15,12,40,12,6,"There's nothing but downside to not sending troops to a given castle - if someone else sends troops, you auto-lose. So that's at least 1 per castle. Hoping other people would send 1, I decided to send 2 per. 80 troops left.

General idea from here: pick my battles.

Castle 10 isn't worth fighting seriously for, but at the same time, other people might agree. In theory, my wide deployment will help against people who focus heavily on high numbers. However, I'd need to win all of 1-7 against someone who takes 8, 9, and 10. I send 6 people there - other people who don't focus seriously on 10 might send a token force, but they might recognize that 10 is worth dedicating a slightly stronger one. In which case they might bump their token force up to 5, like I almost did. That's 4 more to 10, 76 troops left.

I now need to fight it out at 7-9. I'm going to aim to beat the uniform distribution, and the 1-7 1 troop, 7-10 semi-equal troops distributions. To beat the uniform distribution, I'm going to have to have at least 11 in 6-9, so let's go ahead and do that, for 4*9=36 soldiers spent, giving us 40 remaining. I'm going to focus on taking 8 and bump it up to 40 soldiers, in hopes that that will take 8 away from more evenly spread distributions, and use the remaining 11 soldiers to bolster my 2s and 11s against people who are taking similar strategies, leaving me with 3. Bump castle 6 with those, I suppose."
3,3,3,3,3,11,16,21,34,3,
3,3,3,3,3,3,3,3,3,73,"""Clearly, I could not choose the wine in front of you.""
Many semi-optimal subsets use proportional allocations of troops. A configuration which slams troops into a single castle and sends 1 to the others beats many of those. 3 troops to almost all castles beats that variant and its 2 troop ""brother"" strategy.  "
3,2,7,13,5,15,14,12,14,13,I ran simulations with various random troop deployments and this one came out on top in my limited sample.
3,2,7,12,5,18,10,12,13,18,"Excel random number generator matrix calculation. It was a terrible format for this, but fun to figure out. This combination came out as the most frequent winner in a smaller sample than I wanted to test."
3,2,2,2,2,2,2,28,29,28,"* Compete in the three most valuable castles, worth 27 points in total, and hope to win at least one more victory point by forfeit.
* Counter similar strategies by not going all-in on the top three, hedge by covering the remaining 28 points worth of castles with at least 2 soldiers."
3,2,2,2,1,1,1,31,29,28,
3,1,6,5,10,20,10,20,0,25,I shot for more points
3,1,1,11,13,15,1,21,23,11,"I'd like to pick my battles and win those by a little, and if I'm going to lose, lose by a lot. However, I figure some people will send zero troops to some castles, so I'll send one if it could result in an easy win. Otherwise, I just put an increasing number of troops on the castles I choose to fight for. Some numbers are designed to beat some common strategies like all 10's."
3,0,9,0,0,0,21,31,36,0,It doesnt waste troops on castles that I dont need to win
3,0,4,8,0,0,15,35,35,0,Abandon hopes of Castle 10 and put all the eggs in the basket of 7-9 + 4 or 3 and 1
3,0,0,11,0,0,26,27,30,0,I expected that it would allow me to win multiple battles without wasting troops on likely losses.
3,0,0,0,0,0,0,31,32,34,"To win the most wars you need to get >=28 out of 55 points the most often. Giving 30+ troops to each of Castles 8, 9 and 10 will hopefully guarantee you 27 points. Then 3 troops on Castle 1 hopefully gets you that one last point you need."
3,0,0,0,0,0,0,29,32,36,"There are 55 available points, so the winner needs 28.  Castles 8, 9, and 10 provide 29%, 32%, and 36% (respectively) of the 28 points required.  I allocated my troops according to their relative importance, and then put the last 3 on Castle 1 to grab my last needed point."
3,0,0,0,0,0,0,29,32,36,
3,0,0,0,0,0,0,29,32,36,"28 wins, proportional to castle value"
3,0,0,0,0,0,0,29,31,37,"There are 55 points available on the board, so only 28 are needed to win, assuming no ties. I could incorporate ties in my strategy, but I'm an engineer, not a mathematician, it's late on a Friday afternoon, and I'm kind of tired. 28 points can be achieved through winning only four castles: 1, 8, 9, and 10. I concentrated all my forces on those four keeps. I split up my army to assail those keeps with a distribution of 3, 29, 31, and 37 warriors, respectively. I chose those numbers because like a good commander I know my troops. And I know my warriors fight best when arranged in groups of Prime Numbers."
2,18,18,18,18,2,2,10,10,2,"It is advantageous to send at least one troop to a castle, since if your opponent sends zero and then you can win all the points instead of half. I send two in case anyone else uses the one soldier strategy. That way my two will beat their one. I could send three or four but at that point I'm wasting troops that could be sent elsewhere. 
There are 55 total points which will require 28 to win. 
It is my personal opinion that people will either focus on getting the largest numbers to necessary (10, 9, 8 and then 1) to complete 28 points. The smallest numbers necessary (1, 2, 3, 4, 5, 6 and 7) . The best strategy would then be to counteract these two. No matter the strategy, people will focus their forces on the largest numbers they go for. Therefore it is best to stay away from 10 and 7. My strategy will then be to focus 2, 3, 4, 5, 8 and 9 which overshoots 28 points but it is best to have backups. "
2,18,18,2,2,2,18,18,18,2,"I need to win 5 castles and get lucky to win.  I'm guessing people will either try and concentrate their forces on a few castles, in which case my 2 soldiers will win some, or they will spread all their troops out, in which case my 18 soldiers will win some."
2,17,5,8,13,21,34,0,0,0,
2,16,6,15,2,11,21,5,1,21,"Adjusted random numbers. We could say there are three basic strategies (focus on a few high-value castles, focus on many low-value castles, and an even attack). Each of these strategies has an effective counter-strategy, but this naturally requires knowing the opponent's deployment. Random numbers are a form of an even attack, but the variance weakens the counter-strategy. The numbers were slightly adjusted to give advantage when facing an opponent who focused on the 10-9-8-7, without changing the outcome of an opponent who spread evenly."
2,12,13,13,2,2,2,2,26,26,"To win the battle you need to win at least 28 VPs. The combination 10,9,2,3,4 yields exactly 28. Castles 9 and 10 are worth the most therefore they need the highest troop allocations, the rest were split between 2,3,4 evenly while leaving 2 in each remaining castle to win them if they are under defended. This strategy beats common strategies such as: 10 in each, 0,0,...0,25,25,25,25 and 2,4,5,7,9,11,13,15,16,18. Other than that it's mostly a guess at how other players will place their armies! "
2,12,2,15,2,18,2,21,2,24,
2,11,11,11,11,11,3,36,2,2,"Need to win 28/55 points. I'll try to do it by going after the castles others are least likely going to seek out.  Avoid castle 9, and 10, for example. And trade castle 7 for castles 3 and 4, which are equal in value (although I'm spreading across an additional castle).  

Using this strategy, I must win all my castles or the other person wins by default. But trying to win multiple ways is bad - pick a strategy and play it with all your might.  Other players that want to win with the 8 in the least number of castles will require 4 castles - they can't do it in 3.  Since winning with 4 castles requires at least a 9 or a 10, the 4 castle approach would rarely see that player put anything larger than 25 on the 8.  That's the basis for my going above 25 with my 8, but it's too complicated to reveal why I'm going with 36.

If I win, I'll gladly explain my entire strategy, so feel free to reach out to me."
2,10,2,0,20,20,20,20,3,3,"My strategy focuses on the mid value castles and assumes that most people will dump the most soldiers on the high value castles. There are 55 points in total so I need to lock up 28 points to win. Winning castles 5-8 will net me 26 points so I put most of my troops there with a sizeable garrison left at 2 to help secure 28 points. I didn't want to waste resources on 9 and 10, however I left a few soldiers there to keep my enemies honest."
2,8,2,2,16,22,22,22,2,2,seemed like a good way to get 28 of the 55 points.
2,8,2,2,13,18,23,28,2,2,Winning 8-5 and 2 will get 28 points which is enough to win. If someone neglects a castle by placing 1 or less troops I will win that castle high is why I put 2 troops on the other castles.
2,8,2,2,10,18,26,26,3,3,"I wanted to send two troops to each Castle just in case some people ""threw away"" certain castles by sending zero or one troops. Then I competed for the middle castles that would take me to 28 points (8,7,6,5,2). I wanted to send 26 to 8 and 7 to commit heavily to them. I also decided to send 3 to castles 9 and 10 just in case people only sent a few trying to win them. "
2,7,11,14,18,21,25,0,1,1,The race to 28. Trump's America.
2,6,10,14,18,22,26,1,1,0,It seemed like a good idea at the time.
2,6,8,13,18,22,24,3,2,2,"You need 28 points to win, therefore each point is worth 3.5 soldiers. Considering most people that read this site are math and science orientated people and will come up with similar calculation, I stormed the middle (increase the value of 5,6,7) and counter with an attempt to ""steal"" points at the higher end. "
2,6,7,2,20,20,20,20,2,1,"To win, one must achieve 28 points, and so troop deployment reflected this.

Castle 10 was given one soldier, in order to attempt to capture it should an opponent abandon it, but no more as I considered it more likely that opponents would try hard to seize them.

Castle 9 was also given a slightly larger token force for the same reason.

Castle's 5 through 8 were then the lynchpin of my strategy; taking them gave me 26 points, two off victory, and thus I deployed 20 soldiers to each.

I then focused my remaining points on Castle's 2 and 3, with the hope that I would be able to pick up the 2 points necessary at one of those locations.

Nothing fancy, just guesswork."
2,6,2,12,2,18,2,24,2,30,???
2,6,2,2,11,16,21,36,2,2,"Total amount of points available = 55, so we just need to get to 28. I laid out what I thought was a unique way of getting to 28 while still leaving 2 troops on those castles I ""don't need"", as to keep my opponents honest. "
2,5,8,13,20,0,25,27,0,0,get to 28 points
2,5,8,12,17,24,32,0,0,0,"Sum of 1-7 is greater than sum of 8-10, so I'm forgetting those and doing a makeshift exponential function to divvy up the other troops."
2,5,8,11,14,17,20,23,0,0,"I figured everyone else would send too many soldiers to Castles 9 and 10.  I sacrificed those benefits with the hope that I would overwhelm them at most of the other 8.  If successful I would win 36 victory points, at the cost of 19 victory points.

I then roughly allocated the 100 soldiers eight ways proportionally -- sending 3 soldiers per victory point per castle.  This added up to 108, so I took one soldier away from each to return to 100 total soldiers."
2,5,8,11,14,17,20,23,0,0,Ignore castles 9 & 10and try to win everything else
2,5,8,11,13,16,19,22,2,2,The first order solution would be to weight your troops based on 100*(castleNumber)/(totalVP)  My solution beats that by giving up the most contentious castles and then using the above equation on the remaining 8. At Least 2 on each castle to beat out other people who abandon castles.
2,5,7,9,11,13,15,17,20,1,"Give up 10, then spread the rest out, generally weighing by point value without actually doing any math.  Leave 1 in 10 just in case."
2,5,7,1,12,1,1,19,25,27,"The goal is to win 28 or more points.  It basically ""concedes"" 3 castles, allocating only 1 soldier to them*, in order to heavily weight the others.

* The 1 soldier allocations are in case others use a more extreme version of this strategy allocating 0 soldiers to some castles.  In that case, the 1 soldier would get a cheap win."
2,5,6,6,21,0,25,0,35,0,You have the chose if you going for castle 10 or not. The problem with going for it is that how hard you try to win it depends on how hard the other person does. My strategy is to fight harder than normal on castle 7 and 9 and try to win by winning most of the lower castles.
2,5,5,5,11,11,17,17,20,7,hoping to get lucky
2,5,5,5,5,10,16,21,26,5,"Punted on Castle 10 on the whole, added 5 incase somebody tried the same. Distributed down the remaining soldiers"
2,5,0,0,0,21,31,41,0,0,give up the high points castles to win the middle castle points with a small attempt to win the bottom two 
2,4,10,10,15,15,20,20,2,2,
2,4,9,0,0,0,15,15,25,30,"Assume low value castles may be lightly defended, so try to pick up 3 castles for a total of 15 soldiers. Send most resources to highest value castles, and basically hope the archfiend has wasted troops trying to overwhelm me at 4, 5 and 6,"
2,4,8,11,14,16,19,22,2,2,"I want to pick up free points against any strategy that is is only allocating 0 or 1 point to a castle, and I don't want to fight for the two most valuable castles"
2,4,8,10,20,25,25,0,0,0,"Figured folks would go for castles 8,9,10???but 8+9+10=27, and 1+2+...+7=28. If I win the (presumably underlooked) first seven castles, I win the battle."
2,4,7,13,17,17,18,22,0,0,Trying to stack where others don't. 
2,4,7,12,16,19,18,12,7,3,wild guess
2,4,7,9,11,13,19,35,0,0,Give up top two and win everything else
2,4,7,0,16,21,22,26,1,1,"Sounds like an allegory for the Electoral College. The goal isn't to win the most points, but rather to find the most efficient way to 28 points. All troops headed for a losing castle are ""lost votes"" while castles won by more than one troop result in ""wasted votes."" By yielding Castles 9 and 10, you can minimize losses and guarantee wins at castles by assigning a slightly larger amount of troops to castles than what would correlate to the castle's approximate overall value.     "
2,4,6,16,20,24,28,0,0,0,Most people I think would go for the castles worth the most points.  I chose the opposite route - but who knows?
2,4,6,16,1,21,2,22,5,21,Livin' on a prayer.
2,4,6,15,19,23,28,1,1,1,"Assuming most other ranked the castles by placing a ratio of soldiers to victory points, you would end up with 1=1.8, 2=3.6, 3 =5.5, 4=7.3, 5=9.1, 6=10.9, 7=12.7, 8=14.5, 9 =16.4, 10=18.1. Given that there are ways to attack this strategy, assume most people have developed something more strategic. My plan is to start with 1 soldier on each castle so that you claim any castle where someone else put zero. With the remaining 90 soliders, you want to capture at least 23 victory points assuming no ties, keeping in mind the above ratios. If you went after only castles 4-7 you could get 22 victory points by doubling the standard attack and having 10 soldiers left over plus your original 10. This now gives you 15 on 4, 19 on 5, 23 on 6, 27 on 7 and one the other six. Given the other 10 soldiers are likely wasted on the higher numbers, space them out on castles 1-3 proportionately to get you some time breakers, i.e. 13 total solider (initial 3 plus 10) across 6 VPs means 2=1, 4=2, 6=3 with one left over. He will attack 7 just to give the assist on our most important as losing 4-6 could be made up with our tie-breakers in the 1-3."
2,4,6,13,15,18,18,24,0,0,
2,4,6,10,12,14,16,18,18,0,"arranged in with number of troops directly proportional to points, but gave up on number 10 and allocated them to the higher valued ones. figured even if i put 10 on the lower ones, on average not worth it?"
2,4,6,8,12,14,16,18,20,0,"Assuming most people would go for the largest score castle, I focused on trying to win the majority of the lower scored castles by a relatively uniform deployment.  While I'd have to win 1-4 to score as much as the 10, I figure I have to use fewer troops to gain the same number of points.  By alloting only 20 to to 1-4, I believe I have a higher probability of winning those with those fewer points than alloting 20 to the 10 castle."
2,4,6,8,11,14,16,18,19,2,"Sacrifice the top castle, and guard against small numbers at 1 and 10. Linear progression between castles 2-9, emphasizing a little stronger the higher castles."
2,4,6,8,11,13,16,18,22,0,"Try to get opponent to waste troops on castle 10, rough proportionality after that"
2,4,6,8,11,13,16,18,20,2,More troops per castle except minimum effort in Castle 10 as I imagine most people will put a large stake in that one.
2,4,6,8,10,35,0,0,0,35,"Guessing that most would shy away from 10 to take 9-8-7-6 and secure the win. I'm trying to take 10 and 6 and then the bottom to slide past them. I only really need 10-6-5-4-3 to win, but it's better to bank the bottom in case of stranger distributions. "
2,4,6,8,10,14,22,28,4,2,Pretty much just intuition. I chose to mostly ignore Castles 9 and 10 in the expectation that everyone else will over-commit troops there in the hope of increasing my chances to win castles 6-8. 
2,4,6,8,10,14,17,20,9,10,"I started by distributing 55 troops in proportion to the points available. Recognizing that fighting battles at castles 9 and 10 was likely to get expensive, I then distributed another 36 troops in proportion to points at castles 1-8. Recognizing that I may be conceding castles 9 and 10, I took the remaining 9 troops and added them to 6-8 in hopes of capturing those 21 points to offset the 19 I conceded."
2,4,6,8,10,12,16,21,21,0,Castle 10 will be hotly contested.  My foe will deploy many wasted troops there.  I will win many other battles.
2,4,6,8,10,12,14,20,24,0,"A small perturbation of troops proportional to reward, forfeiting the largest castle in favour of the next two largest."
2,4,6,8,10,12,14,16,18,10,I feel like castle 10 isn't worth it
2,4,6,8,10,12,14,16,18,10,
2,4,6,8,10,12,14,16,18,10,2 more than the Castle is worth (which only leaves half for Castle 10)
2,4,6,8,10,12,14,16,18,0,"Each point of the castle worth of 2 soldiers but the last one, since the enemy will try to conquer it by all means. "
2,4,6,8,10,10,12,14,16,18,No particular reason
2,4,6,8,9,11,14,17,29,0,Sacrificed the 10 castle as most people will over value this one.  With 45 potential points then available an equal distribution of the 100 soldiers would put 2.2 soldiers x value of castle on each castle.  I then overweighted the higher value castles some.
2,4,6,8,0,12,14,16,18,20,"I wanted to distribute my soldiers proportionally to each castle value. At two soldiers per castle point, I would need 110 soldiers, so I just dropped #5. Although it reduces my maximum to 52.5, that distribution has the advantage of faring well against very lopsided strategies... I'm guessing :)"
2,4,5,7,11,13,17,19,20,2,"Essentially abandon 10 because it will be largely overshot (win if opponent places 0 or 1 though). From there, place troops according to castle points with some extras on 7-8-9. Will handly beat someone with weighted distribution or too many on 10 which I anticipate will be the most common placements."
2,4,5,7,9,15,17,20,21,0,"I loaded my forces to the mid range castles, figuring that naive opponents would overload castle 10."
2,4,5,7,9,11,13,15,16,18,Closest to the fractional value of each castle.
2,4,5,7,9,11,13,15,16,18,Points were allocated based on the potential points of each castle (100 soldiers/55 potential victory points=1.8 troups per potential VP)
2,4,5,7,9,11,13,15,16,18,"Proportioned based on the Castle's percentage value of overall points available, rounded to the nearest whole number, I'm sure it will be a common answer, but I'm interested in what beats it!"
2,4,5,7,9,11,13,15,16,18,"I used solver in excel to maximize the deployment given the belief that you should put twice as many troops at 2 than 1, and thrice as many at 3 than 1, and so fourth (aha get it?). Lets see if machine can defeat man."
2,4,5,7,9,11,13,15,16,18,Defend each castle with a number of troops equal to total troops scaled to ratio of castle value to total available points. 
2,4,5,7,9,11,13,15,16,18,"There are a total of 55 points, so I divided the forces based on an equal number of troops for each point.  Castle 1 is worth 1/55 of the troops, castle 2 is worth 2/55 of the troops and so forth."
2,4,5,7,9,11,13,15,16,18,"Each castle is worth a certain percentage of the total victory points. I put as many troops there that equal that percentage. If my rival were to put extra troops in any castle, I will get the points for the other castles he left out. If my rival puts the same troops in any single castle, we split the points. 

e.g. the 10 point castle is worth 18.18% of total victory points (10/55) so I put 18 troops there. "
2,4,5,7,9,11,13,15,16,18,Total of 100 troops. The sum of the numbers 1 to 10 is 55. The ratio between troops and total points is 1.8181818181... and so I allocated that many soldiers (rounded to the nearest integer) per point per castle. This should ensure that my troop allocation accurately reflects the potential value of each castle.
2,4,5,7,9,11,13,15,16,18,"Take all the Castle points (55) and divide 100 by that number (1.8181818...) to get the estimated troops per castle.  Multiply that number by the points for that castle to get the number of soldiers for that castle.  This way you distribute the same number of soldiers for the first 4 castles as for the last castle, which are worth the same amount of points.  Because you have to round, you may not get exactly the same answer, but it should be close.  So, you can lose castle 10, as long as you get castles 1-4 to replace the victory points.  So you can even lose the top 3 castles, if you win the other 7.   Or you can win the top 3 castles and the lowest castle and lose the others.  As long as you win some combination of castles that equal 28 victory points, you win, and you don't care which castles those are."
2,4,5,7,9,11,13,15,16,18,multiplied each tower number by (100/55). (55 is sum of tower numbers). Then rounded each number to nearest integer
2,4,5,7,9,11,13,15,16,18,Assigned based on weighted point value of each castle.
2,4,5,7,9,11,13,15,16,18,"Ran 4377 simulations. This placement of troops should win ~80% of the time. The most I found. This is also the placement of troops that equates to the value of each castle. ie C1 get your 2% of the available points, so assign 2% of your troops to it."
2,4,5,7,9,11,13,15,16,18,divide the 100 soldiers by the total # of points 55. allocate soldiers then based on this rate x # of points for the castle
2,4,5,7,9,11,13,15,16,18,Distribution of soldiers based on share of total points each castle is worth.
2,4,5,7,9,11,13,15,16,18,The 100 soldiers are allocated in proportion to the 55 points available.
2,4,5,7,9,11,13,15,16,18,Expected value and some rounding.
2,4,5,7,9,11,13,15,16,18,A proportional distribution forces any opponent to spend the most resources at a rate of 55 soldiers for the majority no matter how it is placed.
2,4,5,7,9,11,13,15,16,18,"55 points are available.  I used a percentage of points for each castle compared to the total.  So Castle 10 gets 10/55, or 18% of the soldiers."
2,4,5,7,9,11,13,15,16,18,I deployed the troops in proportion to the point values of the castle
2,4,5,7,9,11,13,15,16,18,Percentage of castle's points to total number of points (i.e. 10/55=.1818) equals percentage of soldiers that go to that castle (.1818*100=18 soldiers go the castle 10). 
2,4,5,7,9,11,13,15,16,18,"I used the Huntington-Hill method of apportionment, treating the castles as if they were states, as if their point values were residents, and as if the House had 100 Representatives."
2,4,5,7,9,11,13,15,16,18,Proportional to castle worth
2,4,5,7,9,11,13,15,16,18,It's directly proportional to the points each castle is worth.
2,4,5,7,9,11,13,15,16,18,55/100 * number of units to win
2,4,5,7,9,11,13,15,16,18,55/100 * number of units to win
2,4,5,7,9,11,13,15,16,18,"I took the percentage of what each castle contributes towards the goal of 28 points, then used that to come up with an amount for each castle. So Castle 10 gets you 35.7% of the way to the 28 points needed - divided by two and rounded gives you 18. After rounding I had one left over so I threw it in Castle 8. "
2,4,5,7,9,11,13,15,16,18,Simply proportional to value
2,4,5,7,9,11,13,15,16,18,Troops at i = 100 * [Points at i] / [Total Points]
2,4,5,7,9,11,13,15,16,18,Weighted average of what each castle is worth.
2,4,5,7,9,11,13,15,16,18,Close to uniform per point (100 troops into 55 points = 1.8 troops per point).
2,4,5,7,9,11,13,15,16,18,"First of all there's the what I think of as the Electoral College approach.

Well, it's topical.

There are 55 total points available. If one gets 28 he wins. If you put 3 on castle 1,  29 on 8, 32 on 9, and 36 on 10 you'll get there. You're allocating soldiers by castle points. 

But if you lose even the 1 castle, you lose.

Also, this strategy can be defeated by only 10 soldiers. 4 on castle 1, and 1 each on 2-7.

But it would beat my strategy. 

I went for efficiency. 1.8181... soldiers are available per point. If you allocate 14 soldiers to castle 8 there's virtually no waste of soldiers versus points. Approximately zero. It's efficient. 

But you have one soldier left. And where do you put him?

This is where it gets interesting. The best are, If I'm right, castles 10, 3, and 8.

In order, you're wasting 1 soldier for.555.... points for castle 10, or .6 soldiers for about 1 point for castle 3 and 8. 

But people like whole numbers up to six. Then they go for even numbers and multiples of 5. So adding the soldier to the 10 castle is probably not going to help. 

Also it's the least return. 

Between castles 3 and 8 I already have 5 soldiers attacking castle 3. But only 14 attacking castle 8. So I added the extra soldier there and thought that the worst that could happen is that I lose castle 3 and split castle 8 for a net gain of +1."
2,4,5,7,9,11,13,15,16,18,
2,4,5,7,9,11,13,15,16,18,"I used each castles worth out of 55 total points (1/55,2/55,3/55)"
2,4,5,7,9,11,13,15,16,18,Big ballz
2,4,5,7,9,11,13,15,16,18,"Efficiency. 

There are 1.8181... soldiers available per castle point. 

And while the rules don't say that you can't cut up and part out soldiers, I'm assuming that you can't send .6 of one here and .4 there. 

So with my distribution except using 14 instead of 15 on castle 8, there is, on average, no waste. You waste (to one digit) .2 on 1, .4 on 2, .2 on 6, and .4 on 7. 

Then you underspend .4 on 3, .2 on 4, .4 on 8, and.2 on 9.

That's even. And only 99 soldiers. 

But for a minute let's consider the Electoral College approach. (Hey, it's topical.) There are 55 castle points available. 28 wins. So 3 on 1, 29 on 8, 32, on 9, and 36 on 10 is the most efficient to win 28 points. 

And beats my approach. 

But, interestingly, loses to only 10 soldiers. 4 on castle 1 and 1 soldier each for castles 2-7. The EC approach has to win EVERY castle it attacks. With even a tie it loses. 

So my question is what to do with my last lonely soldier? I've already overspent on castles 1,2,6, and 7. Those are out. I've underspent the same largest amount, .4, on castles 3 and 8. So adding my last soldier to one of those gives me the best benefit. But which one?

So now we get to the what is everyone else doing and how do people think territory. 

For 1-5 they use the number they need. For about 6-15 they use even numbers and multiples of 5. Someone might want 12 or 15 of something. Never 13. Or 17. 

Because you're getting into 15 or 20 territory now. 

And no one anywhere ever ordered  or bought 29 of anything. 

So adding to castle 3 moves me from 5-6. I'll take the probable draw because moving castle 8 to 15 soldiers even if I draw is more valuable. 

And moving castle 9 or 10 from 16 and 18 probably won't make a difference. "
2,4,5,7,9,11,13,14,16,19,"Soldiers divided by the available points = (100/55) 1.81
Soldiers were then assigned to castles based on the value of the castle. (99 Soldiers)

(One extra soldier was applied to Castle 10. In case someone chooses the same strategy, then you split the points, so it is not worth applying it to any other than the most valuable one.)

I'm not a big mathematician but i like to try, and i appreciate your riddles :)"
2,4,5,7,9,11,13,14,16,19,"I attempted to pick values equivalent to the victories points value in making up 28 victory points. For example, winning castle 10 gives you 34.7% of the victory points needed to win a majority. The proportional percentage of this is about 18%. (I chose 19 because my math rounding got me 99 soldiers in battle.)"
2,4,5,7,9,11,13,14,16,16,Determine each castle's percentage of the total points then assigned that many units then added the remaining unit to the most valuable castle.
2,4,5,5,10,10,20,40,4,0,Really don't know
2,4,5,5,3,3,4,4,32,38,"The deployment is based on a few key principles:

1. Have a set minimum number of troops for each castle
It pays to leave a few troops in each castle, even the lower valued ones. If an opponent leaves a castle empty (and it is likely that a decent chunk of opponents will do so in order to focus on higher-valued castles) then you can gain substantial points for a very small cost. The minimum is largely trial and error; I ended up with 3 as this was a good compromise between saving troops for larger castles and not getting easily defeated. I did not apply this minimum to Castle 1 as it would not be economic to do so - Castle 1 is worth so little that deploying more than 2 troops to it can quickly become a waste.

2. Fight hard for Castles 9 and 10
This should seem obvious, but if you can win both of these it makes your work a lot easier. On the other hand, if an opponent manages to beat you in these then they have barely any troops left, and thanks to your minimum you have a good chance of winning most of the rest.

2. Pay more attention to Castles 1-4 then 5-6
You can win Castles 1-4 fairly cheaply and get an easy 10 points. If you also win Castles 9 and 10 then you have easily won the war. On the other hand, if an opponent also focuses on 1-4, then the minimum should give you at least a few points in the middle. Combine with #2 and you have a decent shot.

3. Don't try to remain undefeated
No deployment is undefeatable. My deployment, for instance, loses to putting 10 soldiers in each castle, and also to the deployment 1,3,5,7,9,11,13,15,17,19 (in Castles 1-10 respectively) which could be considered the ""mean"" deployment. Other submitters are unlikely to go with these deployments as they are overall quite weak - they are more likely to go with stronger options which nonetheless have vulnerabilities. One likely mistake is to heavily weight 3 Castles instead of 2 - you will not be able to put enough soldiers in each to beat a strong Castle 9 or 10, effectively wasting a larger amount of soldiers while also leaving less for the smaller castles. Remember, any path to victory requires at least 4 castles.

Through these principles and trial and error, I found this deployment to be the most successful, beating 22 other hypothetical deployments."
2,4,4,8,9,10,12,15,17,19,"Not knowing how my fellow Riddler readers will attack this problem, I went for the optimal strategy against a random distribution. The ideal ratio of troops each to point is 1.8181... . I believe my distribution comes as close as possible to achieving that, as measured by the average of castle-by-castle ratios. We'll see how it goes!"
2,4,4,4,4,10,18,20,22,12,"I've played this before! In the past I've seen that 4 is usually enough to contest the lower castles, while committing too much to castle 10 is often a mistake, especially among groups still new to the game."
2,4,2,3,14,21,1,30,21,2,"The best result from a somewhat improved algorithm. The old code is at http://pastebin.com/zT2PifR4, while the new code is at http://pastebin.com/Q3UYquxr."
2,3,11,2,15,18,21,1,26,1,"I wanted a slightly random option that would defeat obvious choices like an even distribution (10 to each of 10 castles). Or going for all the top castles (25 each on 10, 9, 8, x) which I would prevent by winning castle 9. I was initially going to go for all of the lower castles, (7 and below) but that would lose whenever someone spent big on 10, 9, 8, and a random one. My goal here is to win castle 9 almost always, 7 almost always, and usually 6, 5, and 3 without conceding any other castles. Winning these castles would provide 30 points and a win. Losing one of them could be made up if someone sent zero soldiers to castles to some of the other castles. If someone else tried a similar strategy while sending 0 to any castle (especially 8 or 10), that would be a victory for me and a large bonus. "
2,3,8,2,2,0,17,26,38,2,I'm ceding ten because others will deploy a lot of troops there--hopefully many will be wasted.  Trying to get the next three plus Castle 3 which would just barely be a win.  Also hoping to sneak in on some places where the enemy might have put in 0 or 1.
2,3,7,9,13,18,20,24,2,2,"We decided to sacrifice the 9/10, hope that most people will waste too many armies there and we could win the rest."
2,3,7,9,11,13,15,18,21,1,Sacrifice 10 and load up on the rest..look at soldiers per point and give about equal resources per that metric
2,3,6,11,16,26,33,1,1,1,"-Worth leaving 1 point at every castle, in case opponent leaves 0
-People will waste troops on high point castles
-Many people will use round numbers (10, 15, 20), so putting 1 extra point will mean a few free victories.
-The goal isn't to beat the *best* players, but to beat the *most*, and this strat should be decent against a lot of comps"
2,3,6,6,11,16,22,32,1,1,"Don't deploy any substantial troops to 9 or 10; let the others waste troops on them. Focus on 1-8; can still win if lose castle 8, so multiple win conditions. Deploy numbers like 16 and 11 to hurdle opponents who go for multiples of 5. "
2,3,5,10,10,10,10,20,10,10,I figure most people will go heavy on the top two. 
2,3,5,8,13,21,13,21,13,1,Attempt to pick a different strategy than most people
2,3,5,8,12,17,23,30,0,0,
2,3,5,7,10,11,13,14,16,19,"Based on points and number of soldiers, you want about 1.8*points value at each castle. I then rounded to the closest whole number."
2,3,5,7,9,15,17,19,21,2,"Started with distributing by value. 10/55= 18.2% so should have 18 soldiers. Then I figured others will use this strategy so I should take advantage of it. I also wanted to take advantage of those who put too much value on castle 10, but also take advantage of anyone who sends only one soldier."
2,3,5,7,9,11,13,15,17,18,"Figured out the total number of points, and allocated troops based upon each Castle's portion."
2,3,5,7,9,11,12,15,17,19,This is the Banzhaf Power Index value (rounded to the nearest whole number) for each castle. The BPI represents the chance that each castle will be the tipping point that produces victory (number of scenarios in which changing that castle's ownership would change the result divided by total number of scenarios).
2,3,5,7,9,11,12,14,17,20,Math
2,3,5,7,9,11,12,14,16,21,Simulated random troop deployment (with weighted probabilities of being deployed  to each castle) multiple times and this deployment came out as having the highest chance of winning.
2,3,5,7,8,13,14,15,16,17,Weighted average of strategies that try to win 28 points.
2,3,5,7,8,10,12,15,18,20,Logic and skill.
2,3,5,6,7,9,10,30,13,15,
2,3,4,5,11,20,21,30,2,2,"I chose this deployment to basically concede the top two while attempting to score in the lower Castles. I would also beat other people using this strategy because I would take the top two Castles and by varieng the spacing for the others, hopefully tie or score on the lower ones."
2,3,4,5,7,11,20,33,15,0,
2,3,4,5,7,9,11,15,19,25,"I ran a simplistic genetic algorithm, which naturally did not converge but 'floated' around strategies with roughly exponentially increasing men per castle. Intuitively that feels fine, and there is no time to further investigation, as the battle rages on! Must fight now! Or fail. So... I approximated a few of the solutions with a rounded exponential distribution (rate~1.31) and ordered the men to march to their castles. Glory to the winner, honour to the vanquished."
2,3,4,5,6,14,15,16,17,18,Attempting to allocate 1 more than average to each castle.
2,3,4,5,6,14,15,16,17,18,Seemed like a reasonable distribution for exploiting those who go somewhat top-heavy (though admittedly not those who go 25-25-25-25 or 20-20-20-20-20 at the top) as well as those who go for a near-balanced approach.
2,3,3,13,21,22,23,7,3,3,na
2,3,3,5,6,17,18,21,22,3,"Forfeit the obvious top battle but leave some stray troops in case the opponents think the same way. (I would have loved to run some simulations with genetic algorithms, but the weekends are simple a bad time.)"
2,3,3,4,3,3,3,70,3,6,"It takes 28 points to win. Hope that many other strategies will focus on four-tower combinations like 1-8-9-10 and 5-6-8-9. Placing the majority of my army at castle 8 will disrupt many of those combinations. 
Having placed 20+ soldiers at their four targets, my opponent cannot control the remaining towers. Placing 3 of my soldiers at each should be enough to win most of them. If they don't focus on castle 10, they still might increase their allocation there to 5, so bump mine up to 6. 
Castle 1 just isn't worth defending, so I removed 1 soldier. The marginal value of placing soldier #71 at castle 8 seemed pretty small, so I placed this guy where the point value was just starting to feel worth the expense."
2,3,3,3,5,5,7,7,15,50,
2,3,2,13,13,21,2,21,21,2,"I experimented with a few different reasonable placements (and several obvious options like 10 on all, 11 on top 9, 20 on top 5, etc) and this successfully defeated them all. It's a good balance of obvious and contrarian- so a good game theory approach. We'll see how it goes. "
2,3,2,8,3,2,26,25,26,3,"My hope was to get enough points to win (28) through a combination of undervalued castles. I'm assuming people will overfocus on castle 10 (though I put 3 soldiers on it for those who significantly underfocus on it) and overloaded castles 4/7/8/9 to get to 28 points. In case someone went for 25 each on the top 4 castles, I tried to aim for enough to beat out 25 on two of the castles (7/9) and enough to tie and defeat elsewhere on castle 8. On the remaining castles, I tried to put one more soldier than I expected someone strapped for soldiers might put on that castle. Fun Riddler this week. I look forward to seeing if I should pursue a life of warfare or stick to my current day job. "
2,3,2,6,12,3,16,21,31,4,"I wanted to get a leg up on people who chose to deploy based on allocation by overloading castles 9, 8, 7, and 5.  I thought that some might try to capture the top few castles and give up on the bottom castles so I would only need to steal a top three castle to more or less lock up the battle.  I also thought that many would send the largest army to the 10 castle.  I sacrificed castle 10 to shore up other armies hoping that opponents would either overload castle 10 or would send even fewer than I did (unlikely).  I also tried to beat round numbers based on people's affinity for them.  I made a few quirky adjustments (like sending 3 to castle 2 but only 2 to castle 3) just for the sake of randomness."
2,3,2,3,0,11,12,21,23,23,I wanted to maximize my chances at the top end while also giving myself a chance to pick up a few low numbers if anyone decided to completely punt those castles
2,2,18,18,2,18,18,18,2,2,"The total point value of the castles is 55, so a minimum of 28 points is needed to win. I then picked the castles that added up to 28 points or more, while not choosing the more obvious targets (e.g. Castle 10). For the castles that I did not try to capture, I nevertheless assigned 2 troops to each one. Why? Well, if my opponent decided not to try to capture a certain castle at all, 1 troop would be enough to take it. However, it is likely that others are thinking the same thing, so I put 2 for each instead of 1. (Obviously, this number could keep increasing, but 2 was the most I felt I could spare without hurting my chances in my prioritized castles.)"
2,2,14,16,2,18,20,22,2,2,"Focus on getting five medium-sized castles to get just enough to win, then give 2 to the rest (enough to win if the opponent has allocated only 1)"
2,2,11,11,11,11,12,26,12,2,"This beats the 10 on everything spread, also trying to abandon the first two slots with just 1 solider gives me 3 free points. Those who abandon the top slot for a lower spread also gives me the a free 10. Fighting hard on 8 because the 10-9-8-1 strategy needs the 8 to win and dividing it up 25 on each makes the most sense."
2,2,11,11,11,2,2,26,31,2,"Goal is to win 8 and 9 (cede 10 as most valuable) hoping lots of troops end up there, and 3-5 (to get total to 29 points, 11 each to beat any 10s), put 2 troops at least on everything to cover anyone trying to scrimp there (and to cover anyone trying to beat anyone who would just put 1)."
2,2,10,17,5,19,3,1,39,2,"Kind of random, didn't want to put 0 troops anywhere.  Hoping that others over think their strategies."
2,2,10,14,16,22,2,28,2,2,My strategy beats the ponderated troop deployment and gain advantage over strategies that let go some castles
2,2,8,12,14,16,18,20,3,5,"Give up the big castles to win the medium ones, but try to keep enough troops in the big ones to beat other people doing the same thing"
2,2,8,8,18,19,19,20,2,2,Avoid the top - win the rest
2,2,6,12,2,18,24,30,2,2,"Since there are 55 points possible, I set out to try to win 28 points. I picked 5 castles that could get me those points and sent almost everything to those (sending 2 each to the others as a precaution against someone else using the same strategy)."
2,2,6,11,16,2,2,26,2,31,
2,2,6,11,12,14,21,30,1,1,I wanted to protect from the person who did all 10's.  I figured that most people will try to go for castle's 9 and 10.  I wanted some on 9 and 10 because if a person goes for just castles 1-7 which is a winning strategy.
2,2,6,6,6,26,26,26,0,0,Assume most people will concentrate on castles 9 and 10. Try to pick up enough points elsewhere.
2,2,6,6,2,20,20,20,20,2,na
2,2,5,13,2,2,18,24,30,2,To defeat people with similar strategies to mine
2,2,4,11,12,12,12,15,15,15,top heavy with small amounts at the bottom in case opponent put no troops there.
2,2,4,7,12,16,17,1,1,38,"Only need to get 28 pts to win.  If you can predict where most other people will play, then you can avoid putting soldiers there to minimize your losses.  Main goal is 10 + 7 + 6 + 5.  If loss of 10, then possibly get 4 + 3 + 2 + 1 or some other draws.  Can't plan for everything - I'm sure there are others who predict people will put in 10 and they'll move elsewhere.  But if you don't get any of 8/9/10, you need to get everything else.  Average strategy would put 14-15 in spots.  So trying to hedge against that strategy + 10s across.  And aside from those, no idea what others would try, its hard to plan to beat the largest % of people.

I guess you can also think in terms of pts per soldier."
2,2,4,6,10,16,26,30,2,2,"a true commander never gives away his secrets.

ok, fine. i figure that a combination of 1 through 8 will be a majority of points and would secure a win. (1+2+3+4+5+6+7+8 > 10+9)

can't leave the two large castles under-defended though, in case someone tries a sneaky strategy. so send at least 2 to each castle.

use a fibonacci sequence to determine the rest. for the 3rd, send 1st + 2nd. for the 4th, send 2nd + 3rd, etc. if over 100 in total, deduct from the 8th."
2,2,4,5,7,10,10,15,20,25,"Basically set the soldiers for 10, 9, 8. Afterwards distributed the rest in decreasing amounts to 7 to 1. Numbers had to add up and the higher the amount of soldiers for 7-5 the better."
2,2,4,5,7,10,10,15,20,25,"Basically set the soldiers for 10, 9, 8. Afterwards distributed the rest in decreasing amounts to 7 to 1. Numbers had to add up and the higher the amount of soldiers for 7-5 the better."
2,2,4,5,7,10,10,15,20,25,"Basically set the soldiers for 10, 9, 8. Afterwards distributed the rest in decreasing amounts to 7 to 1. Numbers had to add up and the higher the amount of soldiers for 7-5 the better."
2,2,4,4,15,21,22,24,3,3,"I suspect most people will fight heavily for the top two castles. I have decided to punt on these and instead focus heavily on the middle group which is still worth a lot of point, but can probably be secured with fewer troops to each. I did however send a few troops to the top and bottom, in case an enemy decided to ignore them completely, either giving up on them entirely in the case of 9 and 10, or deciding they were not worth the effort in the case of 1 and 2. "
2,2,3,6,8,11,2,2,27,37,"I think people will overemphasize 6-9, so I'm punting 7 and 8 and trying to win the rest."
2,2,3,4,8,14,17,20,29,1,This strategy works well against a random monotonically increasing strategy.
2,2,3,4,7,10,14,22,35,1,Fibonacci series + 1 shifted to account for people overvaluing 10
2,2,3,4,6,8,11,15,21,28,I assumed that an exponential decay function would describe the best distribution.  I wrote a computer script that tried different distributions and select one that was successful.
2,2,3,4,5,15,17,23,27,2,"I initially tried to optimize the number of points per soldier, then readjusted to favor mid to high point castles while maintaining a minimum number of soldiers at every castle."
2,2,3,3,13,21,26,28,1,1,"I expect many people to heavily weight castles 9 and 10. I am going to sacrifice them, except to those who completely ignore them. I also expect many people will also recognize that a main focus will be on the highest castles and will only to send a few troops to the lower number castles. I have 2 and 3 troops to lower castles because I feel like several people will send 1 troop as they anticipate others to ignore these castles. 
My strategy is less about winning individual battles with other distributions and more about winning the war; I'm focusing most of my efforts on the middle numbered castles, with a bias towards the higher castles. I feel like castles 7 and 8 will be the most influential, since they are both part of a ""top down"" or ""bottom up"" strategy. As such, I made sure that both of these castles had many troops. I feel like this strategy is robust against several strategies, hence the appeal"
2,2,2,22,2,2,22,22,22,2,"I identified the minimum number of castles to reach 28 points (4), and placed the maximum number of troops I could while still being able to place 2 everywhere else. I chose 2 everywhere else so that I could win others if someone else decided to put 0 or 1 there, which would make up for losing some of the others."
2,2,2,21,5,3,21,21,21,2,"There needs to be a focus on at least four of the castles in order to enough points to win (28).  Castle 10 is the most likely to draw large numbers of troops due to its high value, so attacking 9, 8, and 7 will hopefully guarantee some values.  The rest of the castles attracted low numbers in the case of an opponent simply not sending any troops or only sending one to pick up any uncontested castles."
2,2,2,21,2,29,2,2,2,36,"focus on 3 castles, disrupt other strategies, capture uncontested castles"
2,2,2,20,2,2,22,22,24,2,"Trying to win castles 9, 8, 7, and 4 to score more than half the points. Also trying to poach castles where my opponent put 1 or 0 soldiers."
2,2,2,16,17,18,19,20,2,2,"we need 27.5 points to get a victory. overloading the top two castles only nets 19 points and i feel like an emphasis will be to get the highest castles. I'm overloading the middle 8,7,6,5,4.  that's 30 points for a victory. the 2 soldiers at the remaing 5 castles are to win castles left with 0 or 1 soldiers and to not completely concede the other 5 castles."
2,2,2,16,2,19,2,15,25,15,"Given that Blotto games are notoriously difficult, I assumed people would not play Nash Equilibrium strategies (this may be a terrible assumption, but I also didn't want to solve for Nash Equilibrium in a different variation of a Blotto game from what I'm used to). With 55 VP total, you need 28 to win. I figured a number of people would try that by going low (1-7), or high (1, 8-10). I thought people would think going high is the obvious choice, and go low. So I mostly went high, but put large allocations on a few small ones other than 1 (since that is needed for both low an high)."
2,2,2,15,17,18,19,20,2,2,"Assuming people would go for the highest value castles the most I started with castle 8, assuming I'd win sometimes, then incremented down, removing one from each lower castle because of the lower priority. Then I dumped 2 into each other castle to catch anyone who tried a similar strategy, assuming they'd only leave 1 to go for the split."
2,2,2,15,16,17,25,5,6,10,Hybridization. Aggressively pursuing lower castle chunks against basic high castle value strategy while leaving medium low numbers to feast on remains of overly clever NYT readership.
2,2,2,14,16,18,20,22,2,2,"Trying yo win all from 4 to 8, which is enough to win the war. Also, trying to win ""free"" castles. "
2,2,2,12,15,18,21,24,2,2,"I decided to try to claim a set of middle value castles that are worth over 100 points, allocating 3 soldiers per point for those castles. The remaining soldiers are distributed 2 each in the remaining castles, in case there are any unguarded easy captures."
2,2,2,12,15,18,21,24,2,2,At least 2 everywhere to win castles against people who leave castles empty/one token soldier.  Focus on mid-high numbers as enough points on offer to get to 28. Do not want to waste a sizeable amount of troop on 9 and 10 has other armies could really focus on winning points there
2,2,2,12,2,17,2,2,27,32,"Focused on securing 29 points, then what would put me over the edge against someone with the same strategy. Attempting to overwhelm my opponent in just enough locations, and pick up easy points at undefended locations (or locations contested by only one soldier, to counter someone using the same strategy). "
2,2,2,12,2,2,19,26,31,2,"Aiming for a sum of 28 (just over half of 55), I'll try to put my eggs in four baskets. Guessing to leave 10 unguarded as a psych tactic against people that put a lot there."
2,2,2,11,16,16,2,2,21,26,"A barbell strategy - i.e. ""all or none"" - felt the most appropriate. Try to win a castle or not waste many soldiers in defeat.  But given that others likely have similar strategies, I left a minimum of 2 soldiers in each, predicting that others will leave only 1. I specifically chose to use numbers like 11, 16, 21, 26 in case my archenemy chose round numbers like 10, 15, 20, 25."
2,2,2,11,2,2,11,15,51,2,"My main tactic is to ignore 10 due to competition, and focus on 7, 8 and 9. I sent a minimum of 2 to the remaining castles, in case they are either unoccupied, or another person had the same thought and sent a minimum of 1 to all the castles. If they also sent 2 as a protective measure, then bravo and we all tie. if 3, well, I hope I won 7 and 8 then, since their forces were spread thin. 4 is bumped to 11 because I wanted to ensure I win against a person that spread 10 across each. Too many possibilities to protect against, I know."
2,2,2,11,0,21,31,31,0,0,"28 points requires 4 or more castles, I wanted to avoid the competition for the top spots as much as possible so chose a wider distribution assuming that fewer opponents would go for the 1/2/3 point castles. This is probably strong against 4 castle solutions (competing strongly for 8 and 7, then challenging at a weaker level for 6 and 4 should hit a weak spot in most 4/5 castle solutions). Probably weakest against larger spreads, and a bit of a crap shoot against other 6/7 castle solutions, just depends where you load up. Fun one, thanks."
2,2,2,10,20,20,20,20,2,2,"I am banking on no one going for low point value castles, so I can grab those incremental points. I placed majority of troops in Castles 4 - 8 in anticipation of enemies going all in on high value castles."
2,2,2,10,13,17,22,28,2,2,Divine inspiration.  
2,2,2,10,13,16,16,35,2,2,Send at least 2 troop to every castle to win over anyone who sends 1 or none.  Put 35 on Castle 8 to counter anyone who focuses on 8-9-10 splitting troops evenly.  Target Castles 4-8 for 30 points out of 55 available.
2,2,2,10,11,16,17,18,11,11,Seems Good
2,2,2,10,2,2,26,26,26,2,"You need at least 4 castles to win, evenly deployed would be 25 each, I'm trying to win those battles by deploying 26.  I expect almost everyone to at least make a token effort for castle 10, so I'm abandoning it.  I need castle 4 on top of my other castles, so the majority of my remaining deployment is there.  I think some people will deploy a single soldier to castles they abandon, so I'm trying to win those battles as well.  This wins against the following logical deployments:

0 0 0 0  0  0  25 25 25 25
16 16 16 17 17 17 17 0 0 0
2 5 8 11 14 17 20 23 0 0 0
0 0 0 0  0  0  21 24 27 31
2 4 6 8  10 11 13 15 17 19"
2,2,2,9,10,25,25,25,0,0,Surrendered higher value castles and focused efforts on several medium sized castles in hopes that they would add to far more.
2,2,2,6,8,16,16,16,16,16,I wanted to distribute them somewhat evenly towards the higher numbers
2,2,2,6,8,12,14,16,20,18,"Modeled chance of winning each castle as an exponential distribution in the number of soldiers used, and then iterated through different possible means (generally increasing as the point value increased, though not always, especially at either end of the scale). Iterated through possible strategies and chose the most successful ones for various values of the means, then cross-tested them for robustness. Achieved about an 80% win rate with this one in my simulations."
2,2,2,5,5,26,26,26,3,3,Never leaving any blank but counting on the middles. 
2,2,2,4,10,12,14,16,18,20,trying to win 4 of the highest 6
2,2,2,3,5,6,9,14,22,35,Picked some Fibonacci like numbers
2,2,2,3,4,6,11,14,22,34,"I noticed that the sum of the first 9 Fibonacci numbers was 88, so if I placed 1 troop in castle 1 and started the sequence on castle 2, I would have enough troops left over to place an additional troop in each castle. This left me with 1 remaining troop for castle 7, as placing 9 troops seems quite inefficient in a game where you expect a lot of 10s. Looking over my answer, it felt a little crazy spending 35 troops on castle 10, so I bumped it down to 34 and placed that remaining troop again in castle 7 so I could beat the 10s instead of tie."
2,2,2,3,3,10,18,19,20,21,Emphasis on winning highest value castles
2,2,2,2,22,22,22,2,2,22,"I never put 0 or 1 soldier in a castle because I suspect that several opposing armies will often put 0 or 1 soldier, and I want to be able to win those.

For some reason, I feel like folks will focus a lot of their troops on castles 8 and 9, so I mostly want to concede those castle, (except I want to win it if the opponents are also conceding the castle).

I also kinda feel like there's not much difference between 13 to 19, but 20, 21, and 22 armies are each more likely to win."
2,2,2,2,21,21,21,21,4,4,"Minimal focus on the very high (people more likely to devote large amounts of troops to capture the most valuable castles) allows me to heavily attack the middle ones.  Capturing 9 and 10 is worth 19 points, but 5-8 is 26.  Putting 4 towards 9 and 10 lets me potentially capture them from other people attempting to metagame similarly to me."
2,2,2,2,18,20,2,22,28,2,"Reduce wasted troops contesting some castles, but still trying to edge out others using the same strategy. Also, making sure to commit heavily to enough castles to get to 28 points"
2,2,2,2,18,18,18,18,18,2,"Everybody will overkill 10, and 1 to 4 are insignifficant. I will put soldiers there anyway, and 2 because many people will choose to put 1 ""for the chance"" in some of the places."
2,2,2,2,16,19,2,25,28,2,"Securing 28 points means securing victory, which can be done with as few as 4 castles. I select the lowest value set of 4 castles meeting this goal to focus on, which I expect to be the least contested. These are castles 9, 8, 6, and 5. I commit 2 soldiers to contest minimal investments in those castles by the opposing warlord, and then distribute the 84 remaining in proportion to the value of my four primary targets."
2,2,2,2,16,17,18,19,20,2,"A lot of people are going to try to win castle 10 and put a lot of troops there. I'm basically ceding it. I put a nominal 2 in case other people are completely abandoning it. 

Anything under 4 isn't worth much (why put people on castles 1-3 when you could put them on Castle 7 and get more points?) Again, I'm putting a nominal 2 to pick up a few easy points against people who abandon them completely. 

I put a high number of troops at castles 5-9. Winning 9,8 and two of 7,6, and 5 is enough to win. Hopefully in the cases where I lose 8 or 9, I pick up a couple points at a lower castles"
2,2,2,2,16,16,16,16,26,2,I basically ignored #10 to focus on 5 to 9. I put at least 2 soldiers at each castle hoping my opponent would just send one.
2,2,2,2,14,16,18,20,22,2,Second submission... for curiousity's sake. No real solid strategy... just go ole gut feeling.
2,2,2,2,14,2,21,22,31,2,"You need 28 points to win.  I went all out on 5,7,8,9.  1 more than I needed but wanted to create some randomness to beat similar strategies.  I then placed 2 on the remaining spots.  If someone beats me on 5,7,8,9 they are likely choosing a similar strategy and 2 on every castle provides some insurance in case I miss one of my key castles."
2,2,2,2,13,24,2,25,26,2,"I focused on a small number of castles that would provide a sufficient number (28) of victory points.  I chose 9,8,6, and 5.  I placed at least 2 soldiers in every castle to defeat anyone who put only 1."
2,2,2,2,11,16,21,21,21,2,"I can win if I secure 5-9, so I focused on those assuming that 10 will be more sought after. I put 2 on each of the others to pick them up if someone only put 0-1 on them. I tried to make sure all of my numbers were not divisible by 10 or 5, because people tend to choose those. I also wanted to make sure 5-9 beat the 10 that would be used in a simple even distribution. "
2,2,2,2,11,11,11,3,3,53,Absolute chaos
2,2,2,2,11,11,2,33,33,2,Aiming for 28 points while defeating what I think will be common strategies. Main weakness will be against something that is even more spread out (beats all the 2s) but also spikes at 5 or 6.
2,2,2,2,11,1,1,23,26,30,High numbers at high value - give up a few mid range castles
2,2,2,2,10,25,20,33,2,2,"Rather than reach the 27.5 needed to win, my strategy is to deny paths to victory for my opponent. Most paths to 27.5 require winning at least one of the castles worth 6, 7, or 8 points. I've also placed 10 on the 5 in an underwhelming effort to block the shotgun approach of 9 troops on each castle. I've very lightly distributed troops to the other castles in the hopes of picking up undefended castles."
2,2,2,2,10,20,20,20,20,2,Concede the 10 pt castle as people will likely fight most for it but leave 2 incase someone wants to concede it. Try to win castle 5-9 and hope that people concede the lower value castles. Keep the 6-9 armies the same to let other people out think themselves
2,2,2,2,10,16,18,21,25,2,"Minimum of 2 per castle as I think a lot of people will send 0 or 1 to some as a man-made strategy, and then enough to the others that I'll come out on top against either an even distribution strategy or an even weighted-by-points one. 

If I had the time I would do some simulations to decide on the best distribution against various strategies, but I have a small child, so such things are not possible. 

Thank you for the interesting challenge - more of these crowd-sourced submissions please! "
2,2,2,2,10,14,18,22,26,2,"Assume folks go after 10 hard, so put minimal support.  Focus on 9 through 5 with varying degrees.  I thought it would be a good strategy to put 1 instead of 0 for the absent castles, but upped to 2 in case others had the same strategy"
2,2,2,2,3,3,4,30,50,2,Hunch.
2,2,2,2,3,2,18,28,39,2,Seeing if I can get some big points and share in some others.
2,2,2,2,2,42,2,2,2,42,"Focus on 2 castles, disrupt others, capture uncontested castles"
2,2,2,2,2,27,33,24,2,2,"I wanted to be able to pick up cheap points by beating anyone who leaves a castle un-attacked or with just one attacker. I figured lots of people might concentrate on winning castles 9 and 10, so I concentrated my forces on three smaller castles that add up to more than 19 points. Varied my troop numbers at the castles 6-8 in case of opponents divining them evenly."
2,2,2,2,2,26,2,2,2,58,"Get easy points, take castles 10 and 6. Relies on opponent leaving a bunch blank or sending 1s"
2,2,2,2,2,22,22,22,24,0,Conceded 10 points hoping to get some of the lower castles in exchange.
2,2,2,2,2,22,22,22,22,2,(i may have mis-typed the first entry... stupid mobile phone. sorry!)
2,2,2,2,2,22,22,22,22,2,"I figured the 10 castle would be highly sought after, so i punted there. I tried to overpower the next 4 castles, as winning those will give me victory. I also put 2 in every other castle, with the idea that some people will punt castles completely, others will put 1 troop in some castles, I will beat the people in those castles which will cover me if they go super heavy in one of my 4 big castles."
2,2,2,2,2,22,22,22,22,2,Revised my last one where I didn't use all of my troops. Math is hard.
2,2,2,2,2,18,18,18,18,18,Top Five (for robustness instead of top 4) and the rest to counter snipes
2,2,2,2,2,17,17,17,20,2,"Fighting a meta of 1s, then dodging a fight for 10"
2,2,2,2,2,14,16,18,20,22,2 men to defeat everyone who just sent 1.  then hope to get lucky with the larger ones.
2,2,2,2,2,12,21,21,22,14,
2,2,2,2,2,11,11,10,38,20,
2,2,2,2,2,9,18,25,28,10,"It seemed prudent to concentrate forces. I did 2 troops to the lower yield castles, just in case lots of people concede those with 0-1 troops (although the case can be made, others used my reasoning as well, which would dictate 3 troops, and on and on). But, I chose 2. Castle 10 is the jewel and people may send a bulk of troops there or either concede it. I don't want to waste troops against players that want it at all costs, but I didn't want to just give up on it either. If a player sends just token forces there, I like my odds with 10 troops. and if not, then I'm glad to let them expend lots of troops against my 10. My thought was to really concentrate forces, and do so at the castles towards the upper half of the castles, with castle 10 being the exception. I need 28 points. My goal is to take a high percentage of the 7-9 castles and hope for a few others. and if the opponent has overwhelmed me at 1 or 2 of those, then I hope my ""beat token troop deployments"" strategy works at enough of the other castles to succeed. Look forward to seeing the data!"
2,2,2,2,2,8,14,28,38,2,Many people will overload on 10 or put 0 or 1 in some castles. I hope to abuse that.
2,2,2,2,2,2,26,29,31,2,Targeting castles 9/8/7. Will try to take any other castles that others allocate 0 or 1 soldiers to with 2 solders.
2,2,2,2,2,2,22,22,22,2,"Because I've run a blotto tournament at my office (and before that, at grad school w/ my students) each summer for the past 10 years, and this strat tends to do well against first-timers. (So maybe not so well against readers of your column but oh wells!)"
2,2,2,2,2,2,20,33,33,2,"Basically I'm sacrificing castle 10 to improve my odds with 7,8 and 9, then hoping the 2 soldiers I send to the other castles is enough to get me 4 more points.

I compared 33 different combinations against each other. This was the best performer overall, and second best when I pitted my top 10 against one another.

It was interesting to see how many of my most ""clever"" ideas would often lose to less sophisticated deployments. A deployment sending 10 soldiers to each castle was 15-17 overall, but 8-2 against my 10 best performing combos.

 My tenth best performing combination is actually 9-0 against the other ones in the top ten. That one sends 2 soldiers each to castles 1-3, 10 to castle 4 and 14 each to castles 5-10. I chose not to submit that one under the assumption that most people would not put as much time and effort into analyzing this as I did.


"
2,2,2,2,2,2,18,21,23,26,"8,9,10 + any other wins. But a strategy that puts all its eggs in that basket is very vulnerable if it loses any of 8 9 or 10. I went heavily on those, but also 7, and put 2 each on all the others to pick up any where the opponent puts 0 or 1."
2,2,2,2,2,2,18,21,23,26,"It wins vs an proportional-value distribution across all 10 castles (1/55 * 100, 2/55 *100, etc); wins against something like 25/25/25/25 across the top four castles as well; and can also win by picking up < 4 of the top castles and stealing cheap points on castles 1-6 if my opponent ignores them completely by going high.  It's weak to a strategy that specifically tries to pick up lots of smaller castles and one big one (which might actually be a winning strat)."
2,2,2,2,2,2,18,2,34,34,Try to edge out people getting 8-10 and pick up a few other castles to reach 27 points.
2,2,2,2,2,2,13,12,31,32,"You need to win 28 points to win.  There are many, many ways to get these 28 points, but it seemed like the most obvious were going for the highest 4 or the lowest 7 castles.  This strategy beats all varieties of both of these that I could come up with.  There are many, many mixed strategies, but this one seemed to have a good balance of spreading out to catch someone not looking at the low castles and focusing to grab some of the high castles."
2,2,2,2,2,2,2,28,29,29,Easiest way to win is get the top 3 castles + 1 other. Put 2 on every other castle to put them all in play too.
2,2,2,2,2,2,2,28,28,30,"Many troops in three castles which are enough to win on their own. Against people doing something similar, hoping to win 1-2 of them and lower castles."
2,2,2,2,2,2,2,21,21,44,"I tried 5 different strategies, pitted them head to head, and this one won more than the others and beat the next best head to head."
2,2,2,2,2,2,2,2,82,2,Guarantee a big number and hopefully pick up some smaller ones where others leave them as zero. 
2,2,2,2,2,2,2,2,42,42,"I think most people will try to get to 28 as efficiently as possible, which requires winning at least four castles with your allotted 100 soldiers. I am banking on overpowering the 10 and the 9 from those strategies and picking up uncontested (or lightly contested) castles to make up the final 9 points. This configuration will beat any configuration that tries to distribute its soldiers evenly (or relatively evenly) between only 4 castles, which I hope many people will do. "
2,2,2,2,2,2,2,2,2,82,Put 2 for everything just to beat everyone who sends one troop just in case someone sends none. Then bet all my marbles on the big guns at castle 10!
2,2,2,2,2,2,2,2,2,2,"Send two to every castle. For each castle, one of the troops stays & attempts to capture, while the other retreats back & reports to me how many troops the opponent sent. With the additional information, I'll have the advantage on how to deploy the 90 troops for a counter attack. Lose the battle, win the war. "
2,2,2,1,2,7,19,19,9,37,"Wrote a genetic algorithm because I thought it would be cute, and let it run for a while. It doesn't converge because it's easy to generate a child that can beat its parents, so to pick a final submission I looked at the best deployment from a few generations and chose the one that appealed to me."
2,1,16,15,16,16,13,19,1,1,"The race to 29 so to speak, if you can guarantee your own total, cede the more valuable castles."
2,1,6,6,7,11,16,21,23,7,"Concede 10 to focus on 9,8,7,6. Avoid round numbers and bid slightly above them. Bid 2 on 1 since most will probably bid 1 or 0 on it"
2,1,5,15,4,13,5,19,14,22,genetic algorithms or something
2,1,4,4,5,11,19,17,13,24,"This deployment was chosen by simulation, based on win% against random allocations, but also beating the allocation achieved by simple weighting. Simulations were repeated until a reasonably high win % was attained. "
2,1,3,7,12,18,26,31,0,0,"A mixture of probability, pop psych, a few borrowed bits from behavioral economics, and a lifetime of table top strategy gaming."
2,1,2,11,16,2,2,2,31,31,
2,1,2,2,5,13,10,12,22,31,"I ran a ton of simulations. I couldn't find a strategy that consistently won, but this arrangement won my mini-tournament. Obviously you need to capture some of the high-value targets but you can't give up on the little ones either."
2,1,1,1,15,2,30,2,44,2,"Focus on three high value castles, and put in token force on others.  I hope is that other players will also have low-commitment castles ( 0 or 1 ) that I can take will minimal commitment.  No statistical backing for this, just hunch."
2,1,1,1,1,1,1,30,31,31,"I'm attempting to ""guarantee"" the 8, 9, and 10 point castles, and then I only need one more point, which I'm hoping to pick up from the ""irrelevant"" one point castle.  Placing 1 point in each castle gives me any freebies.  "
2,1,1,1,1,1,1,30,31,31,kind of just guessing here.
2,1,1,1,1,1,1,27,31,34,"I want at least a chance at each castle if someone deploys no one, and the highest 3 plus the 1 castle are enough to win, so I distributed the remaining troops proportionately to the point values."
2,0,2,0,15,17,20,5,11,28,"I created a spreadsheet that generated a few thousand strategies, each designed to beat the previous strategy by deploying armies in the most efficient way possible to maximize score.  (After winning as many battles as possible by a margin of 1, the remaining soldiers were then distributed by one of three different methods.) I then used a macro to play each of these strategies against every other strategy (War I). I eliminated the weakest strategies (less than 50% win rate) and ran the macro again (War II). I picked the strategy with the highest average War I and War II score."
2,0,0,8,10,12,14,16,18,20,Let's see what happens
2,0,0,0,0,0,0,32,33,33,I need to win the top 3 castles plus one so I tried to optimise for this result.
2,0,0,0,0,0,0,31,33,34,
1,19,1,19,1,19,1,19,1,19,Decided to go for unpredictability while still leaving room for pickups if the opponent tries to go *more* unpredictably; 
1,18,1,18,1,18,1,20,1,21,"Want to win every one that our opponent leaves empty, also want to win evens more than odds because this gets more points"
1,16,16,16,16,16,1,16,1,1,"The total # of points is 55, so you have to gain 28 of them to secure winning a round. It seems prudent only to aim at this minimal amount, forsaking 27.
But instead I send 1 soldier to each castle, in case the opponent stopped reasoning at the previous step.
That leaves me with 90 soldiers to divide. I count on many players fighting mainly for the 10 and 9, but you don't need those to win.
Again, others may think similarly and go for the lowest: 1 till 7.
But we need to take different decisions in order to win, so instead I divide my troops as equally as possible over six castles: 8, 6, 5, 4, 3, 2. 90/6 = 15, so I add those soldiers to the previous one for the selected castles."
1,15,17,14,1,1,11,19,20,1,"Ran a terrible genetic algorithm to find something better than random, then something better than that, etc etc. "
1,11,11,11,11,11,1,41,1,1,"Focus on what I'd assume would be a popular castle, 11s to beat those who use round numbers, and 1s in castles to punish those who leave anything blank."
1,11,1,14,1,20,1,24,1,26,"Should hold its own against top-heavy developments, without ignoring mid- and low-level castles"
1,11,1,1,17,19,23,25,1,1,"Just need to win 28 points, my strategy hopefully takes out people who allocation 10 to each castle and those that try aggressively to take castle 9 and 10."
1,11,1,1,16,19,23,26,1,1,"Trying to win 2+5+6+7+8=28 in as many matchups as possible. Points assigned are proportional to value, except castle two took troops from the others to avoid losing to all 10s."
1,10,12,14,15,20,1,25,1,1,j'ai ??vit?? la 7 (rapport aux pr??c??dents tests de distribution) ainsi que la 10 et la 9 (les plus fournies en points). 
1,10,11,11,11,16,1,26,12,1,Blind luck
1,10,1,1,14,15,20,34,2,2,"Need to capture 28 of 55 points, so ignore 27 points outside of 1 or 2 troop in case someone sends 0 or 1.  Assume people will pursue either a top or bottom strategy, so go for mid range castles.  Also assume people will put more troops behind more valuable castles with either strategy.  Therefore go for castles 5, 6, 7 & 8.  Still need 2 more points to get to 27 so go for castle 2 as well.  Need enough troops in 8 to withstand a 'top' strategy; need enough troops in castle 2 to withstand a 'bottom' strategy."
1,9,10,10,10,10,20,30,0,0,
1,8,2,2,13,18,20,32,2,2,Tried for best chance of getting 28 while losing C9 and C10 assuming other players would spend heavy there. 2 each on C9 and C10 to counter someone else with similar strategy. 
1,8,1,1,18,19,22,30,0,0,"Ran lots of simulations.  I quickly noted that ignoring a castle or two did better, and focusing more on getting the points you need to win made more sense.  Then it became a question of which castles to focus on, and after more tests, came up with these...

I figure most strategies will be top heavy, i.e., focusing on the top 3 castles.  I'm focusing on getting the 28 points I need from the rest.  Go figure.

(The second place strategy ignored castles 1, 6 and 10; 3rd place ignored 8 and 9...  But all seemed to focus on getting the points you need from a minimal set). "
1,8,1,1,14,19,24,30,1,1,I identified 5 castles that would give me enough points to win but had a good chance to be undervalued by others. Then I distributed the troops to those castles in a way that was roughly proportional to their point value and distributed one troop to each of the other castles in the event that someone else's strategy involved leaving a castle undefended.
1,7,11,11,11,11,11,35,1,1,"With 55 points available, the goal is to get 28 points. The most obvious strategy is to concentrate forces into the smallest number of victories: going for 10,9,8 and then one more, or 9,8,7, etc.  The smallest number of victories needed is four, and every single four-victory combo needs to include the 8, 9, or 10.  These castles should see the heaviest fighting, so I want to avoid them as much as possible.  But to ensure my own victory without having to divide my own forces too thin, I need at least one of them.  The 8, as the smallest trophy, makes a logical frontal point of attack, and I have sent enough troops to beat anyone who sends up to a third of their force there.  At the same time, some people will divide their forces up to try and grab as many castles as possible.  My strategy defends against someone sending 10 against each castle, and the single scout armies give me a chance against someone who focuses their forces on a wide selection of weaker castles. "
1,7,8,9,10,11,12,13,14,15,Smooth distribution of troops.
1,7,8,9,10,11,12,13,14,15,"For this, I didn't try to get in the minds of my opponents. I developed a random deployment generator - or tried to. Then pitted each deployment against every other deployment. 

I first selected a random number from 1 to 10. This would be the first castle I would deploy troops to. I selected a random number from 0 to 100 to determine the number of troops. Then I selected another number from 1 to 10. If it was the same as the first draw, I drew again. Then selected the number of troops to send between 0 and 100 less the number at the first castle. Repeat this until all castles have been assigned troops. This produced a lot of identical deployments, which made me question the randomness of this. After further thought, I should chosen the second castle with a random number from 1 to 9. But because I'm up against a deadline, I don't have the time to develop truly random deployments. I can still glean some insights from my faux random deployments.

I got about 10,000 unique deployments and ran them against each other - round robin style. I added in a few of my own (e.g. even distribution, weighted high, weighted low, etc.) I gave 1 point for a win, 0 for a loss and 0.5 for a tie. 

The winning deployment earned 9,663.5 points and was one of my chosen strategies. This beat out the next best strategy (a random one) by almost 40 points. 

Honestly, I'm disappointed in this answer. I was hoping that one of the random ones would prove better. "
1,7,8,9,7,4,11,16,21,16,
1,7,2,11,14,2,2,26,33,2,"I am attempting to beat the strategy that does not have any troops on castles they do not wish to take. I assume those enemies would proportionally distribute troops based upon the win probability share for each castle. I choose the winning strategy of taking castles 9, 8, 5, 4, and 2, while putting down 2 troops on each remaining castle, in case my opponent puts down just 1 or 0 troops. I only put down 1 member of my army on castle 1 as it of such little value that I do not feel it is worth the additional unit.  I figured many opponents would fight for castle 10, or put down a distributional amount on castle 9 and I wanted to ensure I could win a high value castle, so I choose castle 9. An aggressive, distributional player, would put 32 troops on castle 9, thus I put 33. Given I can no longer distribute proportionally, I took the troops away from castles 4 and 5, hoping my counterpart will attack aggressively on those castles, or not at all. Taking castles 9, 8, 5, 4, and 2 like my strategy is designed for, we will battles for me, with 28 points."
1,7,1,1,16,20,24,28,1,1,"Choosing to avoid the obvious targets of 10 and 9, I focused on minimizing the number of numbers I'd need to win to beat an opponent, leaving me with 8, 7, 6, 5 and 2. I then distributed more soldiers to the higher numbers, more or less proportional to their worth. Finally, realizing that some opponents may do the same, I took away a few soldiers and put at least 1 in all of them, to take advantage of any zeroes a opponent may leave."
1,7,1,1,15,15,30,30,0,0,"I tried to find a ""path of least resistance"" to 28 points, choosing to cluster my troops in the mid to upper value castles and leave the highest point castles alone since it would take the most troops (theoretically) to win those points. I also stuck 1 troop in the remaining lower castles to stock pile points in the event that the other player left those castles empty, which would allow me to lose on of the other castles and still win."
1,7,1,1,11,11,28,38,1,1,"Basically give up on 9 and 10 and try to get 5, 6, 7, and 8...and then 2. If I win those all, the other guy can't win. I let a 1 for each of the others in case I oppose some zeroes. "
1,7,1,1,1,1,24,1,30,33,Concentrate forces in an atypical pattern to get to 28 and place one in all other bases if others concentrate forces in only 4-6 bases.  Concentration based in proportion to points at base
1,7,1,1,1,1,1,29,29,29,"I'm assuming a relatively even distribution of soldier deployment across castles to be most common. I'm attempting to ensure victory by taking castles 10, 9, 8 and one other castle. I envision castle 1 being disproportionately targeted, so I am targeting castle 2 to complete my majority, with any other castle completely neglected by an opponent serving as a back up plan."
1,6,8,11,14,17,19,22,1,1,"I'm going with a ""dominate the red states"" strategy and completely abandoning NY and California (castles, 9 & 10).  For castles 1-8, assign them troops proportional to their value.  Finally, take 2 troops from castle 1 and put one a piece in 9, 10 to beat the other ""me"" players out there."
1,6,8,11,1,1,2,2,34,34,"I'm relying on a lot of people trying to win by capturing 4 castles totalling 28 points, and attempting to deny them the possibility of doing this by taking 9 and 10 which are required for any 4-castle winning combination. I figure 5, 6, 7 and 8 will be competitive because they also feature in 4-castle winning combinations so I've instead attempted to sweep up 2, 3 and 4."
1,6,6,6,6,11,11,16,16,21,"I believe most people would set multiples of 5 on the castles so I set the value of my castles slighlty above these multiples e.g. 6,11,16,21. Then I put more soldiers on castles which have higher value. "
1,6,1,13,1,19,1,25,1,32,Minimal commitment to the odd-numbered castles and then distributed the remainder to the even castles based on relative strength of those castles.
1,6,1,1,17,22,22,26,2,2,"I tried to get exactly 28 in a less intuitive way than most people would, and didn't leave any zeroes to ensure I don't give any points away for free."
1,5,5,11,16,21,16,11,7,7,maximize middle points with flat out guessing.
1,5,1,12,1,21,1,26,31,1,"try to win specific sports (just enough for 28 points), take any ""free"" victories"
1,5,1,6,21,26,1,37,1,1,perfect combination of defense and offense
1,5,1,5,1,5,1,5,38,38,I let each soldier choose for himself
1,5,1,1,17,21,24,28,1,1,Attempt to secure 28 points needed to win while exploiting opponents assigning 0s.
1,4,9,3,16,19,7,10,12,19,"This game, at this scale, is not solvable through theory alone.  In a simpler, 3-battlefield case this problem is essentially solved, but with 10 battlefields and 100 soldiers the scale is too large.  

I decided the best course of action would be to optimize over the win/loss rate of a strategy against a set of random possible opposing strategies.  So I booted up R and ran a simulated annealing algorithm to create a strategy that seemed to win a lot.  That failed horribly because the set of all opposing strategies is enormous.  

So instead of trying out random opposing sets, I tried to create a smaller set of ""plausible"" opposing strategies (repeated or nearly-repeated values in 3-10 slots making up the bulk of the allocation, with random allocation to all other slots).  I found a strategy that was robust to my changing various parameters of my ""plausible strategy"" set, and decided to submit that (win rate in my simulator is about 87% across plausible-strategy perturbations).

I don't know if I picked a good plausible strategy set, but that's why I'm submitting my answer! I did check to make sure it beats ""all 10's"" and ""11's and 9's,"" which I suspect will be really common.  I don't beat (0,0,0,0,0,0,25,25,25,25), but that's a risk I'm going to take."
1,4,7,9,15,17,2,19,26,0,"Every strategy can be beaten if known in advance. It's not possible to guess what other people will do, but I've dropped out of competing for castle 10, while trying to ensure I win castle 9 as well as 8, 6 and 5 totalling a winning 28 points. I've spread some remaining troops around the lower numbers in order to hopefully pick up enough points if I lose one of the castles in my primary strategy."
1,4,6,12,15,15,21,24,1,1,Give up on the two highest where opponents will mostly focus their troops. Focus instead on capturing mid-high value castles.
1,4,6,8,12,14,16,18,20,1,"Most people will put some of their troops on the biggest point castle, 10, leaving less for the rest. So I give away castle 10 and divide out the rest according to the castle's worth proportionately. The extras I have I use to gain a slight advantage to higher point castles. I cover castle 10 with 1 troop in case someone else has the same logic as me."
1,4,6,8,10,12,14,16,18,11,Most people will load up the high value so i give away the top spot unless they are spread evenly to try to ensure the next group can be won
1,4,5,5,15,15,15,15,25,0,"No sense fighting over castle 10. Concentrate forces on 9, which is worth almost as much, and the other higher-value castles. A few points to lower-valued castles in case someone concentrated more heavily on the higher-valued ones."
1,4,2,2,3,12,14,16,26,20,"First I picked several intuitive submissions: all 10s, 100 at Castle 10, and ""1, 3, 5, ... 19"", among others. Then I ran a script to pick a deployment that generally performs well against all of them."
1,4,1,11,1,21,1,26,1,33,"I brainstormed a lot of different strategies and played them off against one another. This one (focusing on the even towers) came out on top. 

I tried to look at different ways to get to 28 points, with perhaps some wiggle room. The obvious is just to aim for the high numbers. Towers 8-10 leave you one point short, so you can try to take the #7 or drop all the way to trying to sneak the #1. You can try to abandon the high numbers and just take #1-7. I thought of some sneaky in between answers, like taking towers #4-8. Generally the top heavy strategies worked well. Any strategy that focused more troops on the bigger towers did better than those that divvied out the troops equally.

This final strategy of focusing on the even towers ended up winning the mini battle-royale. I suppose the point is to win the 10, while conceding the 9, win the 8 , while conceding the seven, and so on, netting just enough points to win with a little room to spare. I made sure to get at least one troop to each tower to sneak wins against folks who focus too heavily. I massaged some of the totals to keep them above any product of 5 (11, 21, 26...). I figured 33 was good up top, since most folks won't commit more than 1/3 of their army to any one tower. "
1,4,1,3,11,17,10,21,1,31,Wrote a script to generate a bunch of strategies and play them in a big round robin + elimination tournament.
1,4,1,1,18,21,26,26,1,1,"I'm pretty certain this game isn't solveable and I'm far too lazy to model scenarios so I just played a few practice rounds with ""friends"" and submitted the most successful one."
1,4,1,1,1,1,1,30,30,30,
1,3,6,14,16,17,18,21,2,2,abandon the most valuable castles and attempt to win the mid valued ones
1,3,6,10,15,20,25,15,3,2,
1,3,6,9,1,15,18,21,25,1,Abandon the likely contested Castle to deploy achieve more troops in less likely corseted Castles. Slight weighting to higher castles.
1,3,5,8,13,21,16,13,11,9,"Assumed people will go for high or low values, so I concentrated my higher values toward the middle, with some extra weight going toward the higher value castles."
1,3,5,8,10,13,15,18,15,12,
1,3,5,7,9,12,14,15,16,18,troop PV + overweight middle castles (which may be less defended)
1,3,5,7,9,12,12,14,16,21,"Proportional to the number of points, with a slight variation."
1,3,5,7,9,11,13,15,17,19,It just happens that 2n-1 for n = 1 to 10 is 100.
1,3,5,7,9,11,13,15,17,19,"Some boring weighting based on value, I should probably dirty it up with some noise."
1,3,5,7,9,11,13,15,17,19,"There are 55 total points to be won and 100 soldiers you can use to get those 55 points. Therefore, I have approx 1.8 soldiers to spend, per point. I just multiplied each castle's point value by 1.8 soldiers per point. I then rounded up to the nearest integer for castles 6-10, and rounded down for castles 1-5, since I'd rather win the better castles and I'm more okay with losing the weaker castles.
This had the interesting side effect of my troop deployments merely being equal to the consecutive odd integers from 1-19."
1,3,5,7,9,11,13,15,17,19,"Thinking economically, you want to distribute your resources efficiently.  Efficiently in this case means beating your enemy by 1.  With that in mind, I've aimed to narrowly win as many castles as possible.  I'd love to see how other simple strategies (e.g. 20 on each of the top 5 castles) would perform."
1,3,5,7,9,11,13,15,17,19,"I determined that since there are 55 available points and 100 available troops, each point was worth approximately 1.82 troops so I attempted to distribute the troops based on that. "
1,3,5,7,9,11,13,15,17,19,"I wanted to have my greatest chance of winning to be at castle 10, but to leave those that totaled up to 10 a similar probability.  So castle 10 has 19 soldiers, but 9 and 1 combined have 18 soldiers, as does 8 and 2, 7 and 3, and 6 and 4."
1,3,5,7,9,11,13,15,17,19,"Scale defense to value of castle divided by total points, with bias to top castles. "
1,3,5,7,9,11,13,15,17,19,soldiers are roughly proportional to victory points
1,3,5,7,9,11,13,15,17,19,"Sent troops that roughly equate to the value of each castle, and hope it works out in the averages."
1,3,5,7,9,11,13,15,17,19,Made the % of troops near % of points available rounding down. Put extra troops on biggest one evenly spread.
1,3,5,7,9,11,13,15,17,19,proportionally to the Banzhaf power index
1,3,5,7,9,11,13,15,17,19,"Strict linear distribution, using 100 as the total, and trying to maximize Castle 10 without losing all the other ones.  Started at 1 troop for Castle 1, then kept adding until I filled all of the 100 spots."
1,3,5,7,9,11,13,15,17,19,A simple pattern. 
1,3,5,7,9,11,13,15,17,19,"I felt like they should be distributed mostly proportionally to the amount of points up for grabs, and so 2p-1=s fit that idea."
1,3,5,7,9,11,13,15,17,19,I decided to choose soldiers proportional to the values of each victory point.
1,3,5,7,9,11,13,15,17,19,I used each castle's Shapely-Shubik power index to distribute troops.
1,3,5,7,9,11,13,15,17,19,basically deployed 1.8 per castle point rounded down
1,3,5,7,9,11,13,15,17,19,"Interestingly, my naive solution was to just use an arithmetic sequence starting at 1 with an increment of 2. My long winded solution below comes out as almost the exact same result, so I just went with the naive one.

After much back and forth, I gave up on trying to predict others' strategies and assigned soldiers solely based on my perceived relative importance of the castles for victory. I used the following python script to determine relative importance of castles. (I ignored ties in my analysis to keep it simple)

from collections import defaultdict
from itertools import combinations
from math import ceil

# constants
SOLDIERS = 100
CASTLES = 10

total_points = sum(range(CASTLES + 1))
victory_points = ceil(total_points / 2)

# 1024 (2 ** 10) combinations of conquering 10 castles
combs = []
for x in range(11):
    combs.extend(list(combinations(range(1, CASTLES + 1), x)))

# Half the combinations have enough points for victory
winning_combs = [comb for comb in combs if sum(comb) >= victory_points]

# Weigh importance of castle by inverse of total castles needed.
# Only count a castle if victory depended on it.
weighted_score = defaultdict(float)
for comb in winning_combs:
    total = sum(comb)
    weight = 1 / len(comb)
    for val in comb:
        if total - val < victory_points:
            weighted_score[val] += weight
            
# Assign soldiers proportional to weighted score
normalizer = sum(weighted_score.values()) / SOLDIERS
soldier_proportions = {k: v / normalizer for k, v in weighted_score.items()}
"
1,3,5,7,9,11,13,15,17,19,
1,3,5,7,9,11,13,15,17,19,Roughly proportional to the value of each castle
1,3,5,7,9,11,13,15,17,19,linear progressions sound good
1,3,5,7,9,11,13,15,17,19,Nothing more clever than trying harder for the high-value castles.
1,3,5,7,9,11,13,15,17,19,Even distribution weighted by point value
1,3,5,7,9,10,12,14,21,18,"The total number of possible points is 55. You are given exactly 100 soldiers to battle with, meaning that you should spend on average around 1.8 soldiers per possible victory point. If you multiply 1.8 times 1, 2,3,4... etc: You get 99 points, 95 if you rounded down. Then I just distributed them based off of this and put the extra 5 on Castle 9. "
1,3,5,1,10,12,14,16,18,20,"Even, weighted distribution, but writes off castle 4/adds 1 to 5-10 to try to beat anyone with the same idea/distribution.  "
1,3,4,15,16,20,20,20,0,1,Played against a coworker and this seemed like the most optimal.
1,3,4,8,12,15,18,19,20,0,deploy none at 10 because it would increase odds at several other castles
1,3,4,6,9,11,14,16,17,19,"No very tricky strategy here, I simply assigned the soldiers according to the value of the castles (e.g. a castle worth twice as much should get twice as many soldiers), using the formula S(n) = round( n*100/55) (55 being the total value of all the castles). Then, I removed one soldier from castles 1-4, and added one soldier to castles 7-10, specifically to crush the mirror strategy."
1,3,4,6,8,12,14,16,17,19,
1,3,4,6,8,10,13,19,36,0,"Other folks will load up most on 10, then 9, then 8. By ignoring 10, I have a greater chance of winner the others."
1,3,3,9,12,19,16,17,20,0,"Simulations. I isolated a couple of common strategies that I would expect people to use, generated a thousand variations of each of those strategies, posted those soldier in their castles, and then threw thousands of random combinations of troops at the castles to see which combination did best on average. I have no idea if it will work, but hypothetically it should perform reasonably well.  "
1,3,3,3,3,15,2,34,34,2,
1,3,2,4,8,20,26,34,1,1,"Avoid 9 and 10, defeat distributions like 34-33-33 and 25-25-25-25"
1,3,1,1,20,21,21,30,1,1,"7 and 8 are the most critical castles.  Between 4 and 7 castles are needed in total (1-7 to 1+8-10).  By concentrating on 5-8, it concentrates forces and may benefit from any abandoned castles fights."
1,3,1,1,17,20,21,30,3,3,"It takes 28 points to win. At first glance, it seems beneficial to take over castle 10, but that doesn't even get you halfway there. The simplest answer would be to control Castles 10, 9, 8 and 1, while placing most of your troops in castles 10,9 and 8. Having 30 troops in castle 8 should be enough to control that, and then trickled the remaining troops among 5, 6 and 7 hoping these are mostly ignored. That still leaves me with 2 points needed to win. I place 3 troops at castle 2 since most people will only place 1 there. Those trying to be cheeky would place 2, but they still lose when I place 3. I used the same strategy for castles 9 and 10 as a fail safe in case one of my ""key"" castles are not conquered."
1,2,11,6,11,2,13,21,31,2,"My first thought was that I wanted to win the castles with the highest four point totals, put nothing on the lower six.  This would give 34 points and a win by a good margin.  However, I would have to win all four of these to win, and it was likely that many opponents would put a larger number of soldiers on any one of these to steal one from me preventing a win.  Since only 28 points are needed to win, I could remove a all of the soldiers from castle #7 and place even more soldiers on castles 8, 9, and 10, and just put a few on castle #1 to hopefully win by a smaller margin.  However, I found that an opponent that put 10 on each would win against this strategy.  I considered a strategy that would win me castles 1 through 7 (also totaling 28 points) but that started spreading my troops too thin making it likely that an opponent would place more than me on just one castle which would cause me to lose.  I started to realize that it is not wise to put zero soldiers at a castle; even putting just one soldier at a castle could help me win in cases where my opponent left castles unattended (which I figured would happen often).  However, I figured many people might realize this as well so I wanted to put two on such a castle if possible.  I also figured that many opponents would go for castle 10, since it is the highest valued, so I shifted my strategy to lower castles.  I also figured people would tend to put ???nice?? numbers on castles such as multiples of 5 (or a dozen) so I tried to exceed those numbers by 1 soldier (putting 6 on a castle instead of 5, 13 on a castle instead of 12 and 21 on a castle instead of 20).

I had some of my students submit solutions, and saw how my various solutions did against them.  Obviously I couldn??t satisfy all of my criteria, so I went with one I liked.  
"
1,2,6,0,5,9,21,28,28,0,"I ran a lot of simulations and couldn't find anything which was always optimal (not to mention any strategy can be beaten if you know it). This one seemed to do well in several simulations, and emphasizes a strategy which avoids trying to win castle 10, but tries really hard to win 7, 8, 9."
1,2,5,10,17,23,20,13,6,3,I started with a normal distribution and skewed it left (by eyeball) to center over castles 6 and 7 because that felt right.
1,2,5,9,14,19,21,17,12,0,"
No idea why, but I have a good feeling about using a Poisson distribution with lambda = 1/e*10, solving the pdf for 0:9, and flipping the array,
 such that pdf(0)= proportion of troops at castle 10, etc.

I then assumed I would never win # 10 with only 3 soldiers, so I added them to castle 9."
1,2,4,14,2,2,2,36,34,3,Trying to counter the counter counter meta.
1,2,4,11,12,17,22,29,1,1,"At least one soldier on each castle, just in case they are not covered by the opponents.  Furthermore, we don't aim for castles 9 and 10, because they are likely popular.  Instead, we go for castles 1 - 8, for a total of 36 points.  This means we can loose any *one* of these castles (including castle 8), and still win.

The numbers are chosen to protect against a number of strategies."
1,2,4,8,2,4,8,16,32,23,
1,2,4,6,9,12,16,21,25,4,no particular reason
1,2,4,6,9,11,13,16,18,20,"First, it is easy to show that no pure strategy is a Nash equilibrium.  For any given strategy, it can be beaten by a strategy that has the same deployment except that 9 soldiers are taken from the castle with the most soldiers and distributed equally to all of the other castles.  Finding an optimal mixed strategy is computationally intractible, and not allowed here anyways.  

I looked at optimal strategies for smaller versions of the game.  For 9 soldiers and 3 castles, [0,0,9] is an optimal strategy.  For 16 soldiers and 4 castles it is already difficult to find an exact solution.  Solving it approximately by using the fictitious play method (https://en.wikipedia.org/wiki/Fictitious_play), shows higher coefficients placed on strategies with more soldiers on the higher points castles, but not all on the highest point castle.  This makes intuitive sense because the higher points castles are more valuable, but you don't want to put all your eggs in one basket because the other castles combined are worth more than any one castle.

Therefore I chose a strategy where deployments to each castle were approximately proportional to the point value of the castle.  This is as reasonable a strategy as any, given that no pure strategy is optimal."
1,2,4,6,9,11,13,15,18,21,average of 1.81.. soldiers per point with some weighting to the higher point castles away from lower point castles.
1,2,3,18,17,16,15,14,4,10,I figured that most people would stack their top castles.  I also wanted to pick something that would beat an even 10 across.  Getting 5 Wins and a Tie is easier than 6 wins.
1,2,3,6,9,10,13,16,18,22,"Again, I used a simulation of between 1000 and 2000 players, some attempting to play optimally, some attempting to play randomly, and some a hybrid of the two.  I found that, the more optimal players, the more the optimal distribution steadily increased from 1 to 10.  The distribution I chose is a compromise between many simulation parameters."
1,2,3,5,8,12,10,15,18,25,I have faith that I can win submitting only 99 soldiers.
1,2,3,5,7,11,15,19,23,14,"Tested strategies against random generated values in a monte carlo. Exponentially weighted distribution worked best. From here, I took points from castle 10 to add 1 to 1,2,3,4,5 and 2 to and 6,7,8,9. The logic in it is that I expect many people to arrive at the exponentially weighted distribution, and I only need to win 6, 7, 8, and 9 to beat them. "
1,2,3,5,7,10,14,16,21,21,"I started with deploying troops in proportion to the marginal value of winning that battle. For Castle k bid the closest integer to 20k/11. That maximizes the expected number of points but doesn't necessarily maximize my winning percentage. So I simulated the bidding strategies of others 10,000 times from a beta distribution and noticed that small adjustments to my original deployment could lead to improved results."
1,2,3,5,7,9,15,17,19,22,Chaos
1,2,3,5,6,13,15,17,18,20,figured most common strategy would be i/55*100
1,2,3,4,10,20,20,20,20,0,Sacrifice Castle 10 in hopes of picking up more lower-level castles
1,2,3,4,10,12,14,16,18,20,Allocated troops to all castles (didn't want to give up any without a fight) in proportion to their points to start.  Allocated the remaining by doubling allocation from the top down until troops were all deployed.
1,2,3,4,10,12,14,16,18,20,"Even soldier per victory point leaves 45 remaining, then double that up, starting at the highest value castle. We run out of soldiers for the four lowest value castles."
1,2,3,4,10,12,14,16,18,20,"Without knowing any game theory, it seemed to make sense to send soldiers roughly proportional to how many points each castle is worth, and I was too lazy to get them properly proportioned!"
1,2,3,4,8,10,15,22,25,10,"Sacrifice most of the castle 10 point victories by only placing a 10 point value on castle 10, but hopefully claiming many of 9,8,7,6 castles off of anyone who devalued them."
1,2,3,4,7,13,17,23,27,3,
1,2,3,4,6,16,25,33,4,6,"Wrote a simulation to choose semi-random strategies and run iterative tournaments to find the best-of-the-best. Most winning sets were monotonic, but I like the idiosyncrasy of this one."
1,2,3,4,5,10,15,20,40,0,"The basic strategy would be to devote more troops to higher value castles, but if everyone does that then it comes down to the specifics of the ""ramp."" Hopefully by sacrificing castle 10, and spreading those resources between 7, 8, and 9 I will gain a relative point bonus of 14."
1,2,3,4,5,6,15,30,33,1,"Knowing many wouldn't send any to multiple castles, I wanted to bring at least 1 to all. I also figure many would focus on winning #10 I focused on winning 6,7,8 and 9. "
1,2,3,4,5,6,7,8,9,55,"Put a lot at 10. Distribute enough at each other one to hopefully steal some matchups while leaving me lots for #10. It might be an awful strategy, but let's see what happens"
1,2,3,3,4,16,21,24,24,2,"click peck number, peck some more numbers, min 3, then try to win 7 then sort them; ignore #10"
1,2,2,10,20,21,16,21,5,2,
1,2,2,6,11,11,16,21,26,4,"I am putting more soldiers in higher castles, but not in castle 10, which I expect many players will put a large number in, and which would thus be too expensive to contest.  I also expect that most numbers chosen will be either 1 or a round number, and thus my numbers are almost all either 2 or one more than a multiple of 5.  I put 4 in castle 10 rather than 3 to increase my chance of winning against others who don't take the castle seriously, at the cost of putting 1 in castle 1 rather than 2 because even winning it with 2 isn't worth much."
1,2,2,6,10,21,22,32,2,2,xD
1,2,2,5,10,5,21,26,27,1,"First, I wanted to make sure that I contested each castle, in case an enemy chose to send no soldiers or only 1 soldier, to prevent strategies like what I am planning from dominating. Second, I'm guessing that most contestants will pour a lot of resources into Castle 10, and that Castle 10 will receive a disproportionate share of most defense. By essentially forfeiting Castle 10, I can reallocate those resources to winning Castles 7-9, which make up almost 44% of the points available and means I only need to win 4 other points somewhere else. Under the assumption that I either win all 3 of those, I only need to win 4, 5, 6 or 2 and 3 to win; under the assumption that I win 2/3 of Castles 7-9, then I need to win Castles 4-6 or 2/3 of those and both of Castles 2-3. But unless the opponent only sent a small number of soldiers to Castle 10, they won't be able to effectively contest all of the remaining castles and still win one of Castles 7-9 (unless they only sent 0 or 1 soldiers to the ones they lost, in which case their strategy effectively counters mine).

I'm hoping that this will generally dominate the strategy of ""30 or more to Castle 10, 15-24 to Castles 7-9, almost nothing to the remaining castles"" as well as the strategy of proportional distribution based on point value (18 to Castle 10, 16 to 9, 15 to 8, 13 to 7, 11 to 6, etc)."
1,2,2,3,16,17,18,19,20,2,Give up castle 10 and concentrate on castle 9 to 5 in the hope that most will try to concentrate on castle 10.
1,2,2,3,6,13,16,18,19,20,"It appears there are 109 choose 9 ways to place 100 interchangeable soldiers in 10 distinct castles; this is about 4.26 trillion (4.2E12) ways; each possibility can play against another, so there are about 1.82E25 total possible games, too many to evaluate.  Smaller scale simulations (up to 40 soldiers in 5 or 6 castles)showed little advantage for the low-value castles, while there were much better wining percentages (with a flatter distribution of soldiers) with the higher-value castles.  At least 4 castles would need to be populated with soldiers to have any chance of winning."
1,2,2,2,20,20,2,25,25,1,
1,2,2,2,16,21,0,26,29,1,"Prioritizing castles 5, 6, 8, and 9 concentrates my forces on securing exactly the 28 minimum points required to win, while avoiding wasting forces on a massive arms race at Castle 10 and, to a lesser extent, Castle 7. Leaving 1 or 2 soldiers at most of the other Castles allows for some flexibility, since I can afford to lose 1 or 2 of my prioritized castles if the opponent ignores some of the other castles."
1,2,2,2,11,15,30,2,2,33,"I figured that some people would focus on castles 7-10, because you can win with just them, and that others would focus on 1-7, because they also give enough points to win. The people who aim for 7-10 will be beaten by my strategy because they will most likely lose 7 and 10, or at least be tied. People who aim for 1-7 will almost certainly lose 7, and perhaps tie for 6. People who put 10 in every castle will lose 5, 6, 7, and 10, which makes me get 28. People who allocate their soldiers like 1-3-5-7-9-11-13-15-17-19 will also lose 5, 6, 7, and 10."
1,2,2,2,7,9,18,27,30,0,"By avoiding the 10 value - I both nearly guarantee the next 17 points for myself, and can view any investment my opponent makes to the 10 value a wasted effort (or lost soldiers) and giving me a numerical advantage for the remaining points.  They may actually only get 10 total."
1,2,2,2,5,15,16,23,32,2,I made the game and played around with it on http://www.solidmecha.com/game/CastleCalc/
1,2,2,2,2,29,30,30,1,1,"I think if I can secure 6, 7, 8 plus some of 1-5 I will beat out those who go all in on 8, 9, 10."
1,2,2,2,2,5,11,15,24,36,Loading up on the top makes the most sense.
1,2,2,2,2,3,3,23,31,31,"I don't know, there's no right Blotto game strategy. I wanted to win the most valuable castles as often as possible without giving the lesser castles away for free."
1,2,2,2,0,14,16,18,21,24,
1,2,2,1,16,22,26,27,1,2,Overthought and undermathed
1,2,2,1,1,23,23,23,23,1,Winning towers 6-9 will yield a point value of 30 points out of the possible 55 which would win. In order to divide them I sent one to the other remaining towers which left me with 94 troops for four towers. 94/4= 23.5 so 23 troops per tower. The remaining two troops were placed on  tower 2 & 3 in case of a tie. 
1,1,29,24,19,1,12,11,1,1,"I wrote a script to compare a bunch of what I though were reasonable strategies, and this one performed the second best compared to that set of plans (the best strategy used 0's instead of 1's on the non-priority castles, but I figured this one was probably more robust against crazy strategies other people might use that have a bunch of 0's).  You can find my script here: https://github.com/jakewalker56/ml-lab/blob/master/visualization/riddler_castles.R "
1,1,19,1,12,19,19,26,1,1,"Aiming to grab a ""non-optimal"" >28 using numbers that add to 30 (wasting soldiers) and splitting extras for ties and a change to grab undefended castles. Boosted chance on 8 to stop losing to a 25/25/25/25 split on the top half. Took the extra soldiers from 5 since it doesn't feature in many easy 28 additions"
1,1,17,18,1,19,20,21,1,1,"I didn't want to put too many resources into the highest values, assuming others will put plenty of resources into them. By ceding those, I have more soldiers left for the others. Similarly, I assume some people will have this thought and put lots into the lowest values, or think both of those things and focus in the center. My distribution avoids the bottom (1 and 2), the top (9 and 10), and the very middle (5). The castles I have put resources into total exactly 28 points, which is exactly the amount necessary to win. If I do take all five of those, then I have no need for any other castles. I then sent 95 of my soldiers to these five, weighted ever so slightly to the higher value ones. This leaves me five left to send, one each, to the other five castles, so that if others leave any castles empty I can grab the points almost for free, potentially saving me if I lose or tie one of the five I??m focusing on."
1,1,16,16,16,1,16,1,16,16,"I'm trying to win castles 3,4,5,7,9,10.  If I win any five of the six, I'll win."
1,1,16,16,16,1,16,1,16,16,"I'm trying to win castles 3,4,5,7,9,10.  If I win any five of the six, I'll win."
1,1,15,17,1,21,21,21,1,1,"Pick 5 castles to hold most troops that sufficient to win if I get them all.  1 instead of 0 in others to pick up easy wins if somebody puts 0.  Avoid 9 and 10 as the most valuable, where many people will put many troops.  "
1,1,13,1,16,21,21,24,1,1,"This is a game to 28, avoiding the most obvious big wins, and going for the middle ones should optimize my wins in the middle.  Avoiding multiples of 10 and 5 should reduce ties.  Keeping one in each castle will pick those up for people who completely ignore those"
1,1,13,1,1,1,15,31,1,35,"I rolled a 10-sided die to choose which castles to focus on - until I got a set that got me 28 points (the minimum to win) in the minimum number of castles (4). These castles would be my focus where almost all my troops would go. This exploits strategies that try to win all or almost all the castles.
Putting at least 1 in all castles is attempting to exploit strategies that don't send any troops to certain castles (I initially tried a strategy where I was exploiting this by putting at least 2 in all castles, but it seemed this didn't leave enough for the ones I was focusing on).
I divided my remaining 94 soldiers into the castles I was targeting by proportion of points they were worth. I made a few adjustments to that in order to try to exploit strategies which divided proportionally (and to avoid similar exploitation myself). This is also to try to make up for the loss of 6 soldiers to my previous point.
I'm sure this loses to computer-simulation-designed strategies, but I think it will do well against a large range of 30-minutes-of-thinking strategies."
1,1,11,12,1,19,24,29,1,1,Derp.  I submitted the opposite order 2 minutes ago (along with my explanation) because I used a different naming convention in my code.
1,1,11,11,1,21,26,26,1,1,"Target smallest, lowest 5-castle win. "
1,1,11,11,1,11,11,26,1,26,
1,1,9,14,1,1,20,25,27,1,"my first thought was to distribute troops optimally by thinking about a ratio of 100 troops to 28 victory points needed. Then thinking that many others would do this and look for opportunities to round up. I thought I would go contrarian and look for towers where people might round down and use that along with trying to pick high value towers. That led me to towers 3,4,,7,8, and 9. I distributed my towers there with heavier weighting towards towers with high value and higher possibility of others rounding down."
1,1,9,1,1,18,20,23,25,1,Focus on 5 castles to get to 33 points
1,1,8,8,20,20,20,20,1,1,Try to win mid level castles
1,1,8,1,1,15,24,24,24,1,"Assume everyone will load up on 10, so don't place troops there (put one just in case someone else thinks the same way and puts zero there).  Try to win 3 of the four with a high percentage, and half the other 2 to get to 28 VPs."
1,1,7,15,17,19,19,19,1,1,To win...
1,1,6,11,16,16,21,16,6,6,"Anticipating overweight attacks at the extremes.  Tendency to round to nearest five gives opportunity to beat those ties by one, maximizing that soldier's value.  Adding a bit extra to seven as a hedge against favoring an arbitrarily luck number.  Never zero a number as a nearly free win with one solder can make a difference."
1,1,6,11,15,1,17,21,26,1,"castle 10, 6, 2, and 1 are not the objective. I put 1 so that when I face someone who put 0 I will win, but I expect to lose them. The rest is a little random but tries to maximize chances against an opponent who chose a monotonically increasing function."
1,1,6,11,11,16,21,27,3,3,Don't waste troops on 9 and 10
1,1,6,10,20,20,20,20,1,1,
1,1,6,1,1,1,1,24,24,40,"A top heavy strategy, hoping more people shoot for the middle, and then grab a couple of points extra if someone puts 0 for a group."
1,1,4,13,14,15,16,17,18,1,I sent at least one dude to each castle to take any that someone might just decide to forfeit.  I mostly forfeited the most value castle and then distributed the rest in such a way i thought was likely to take castles from other strategies.
1,1,4,11,13,21,12,26,8,3,"To guard against as many possible ""simple"" strategies as possible (e.g. placing 20 on each of the 5 highest castles etc.)"
1,1,4,10,16,22,22,22,1,1,"I think most people will go for 9 and 10, so I put most of my troops at 6, 7, and 8"
1,1,4,10,12,15,17,19,21,0,"I am conceding 10, because I think people will overload that castle. In distributing the troops that would have been there, I think I have a great chance of winning castles 4-9, essentially sealing a victory for myself."
1,1,4,8,10,17,24,25,5,5,"one shouldn't throw away the small castles completely, or overbank on the 10 point castle.  Even the 9 point castle is a risky move.  The opponent is likely to waste large numbers of forces on the biggest prizes, leaving the high to middle values untaken.  There are more points by a significant margin in the 4-8 range than in 9 and 10.  and at least some opponents won't spend anything on 1-3.  thus, having one person in each of those wins at least a few battles."
1,1,4,6,8,10,14,16,18,22,I'll maybe win 1 or 2 points from sending one soldier to castles 1 and 2. But then I figured adding more soldiers gradually to the remaining higher-point castles would win me castles worth bigger points. 
1,1,3,6,8,9,10,16,35,11,"Deployments that are evenly distributed or evenly weighted by victory point value lose to any distribution that ignores some castles to concentrate on winning a specific combination worth more than half the VPs.

Ignoring some castles to focus on others requires winning at least four castles, and this most focused case requires winning at least one of the ninth and tenth castles.

I chose this strategy to defeat four-castle focused strategies that rely on winning castle 9 yet remain strong against hybrid weighted-focus strategies designed to beat an even distribution."
1,1,3,5,12,15,20,18,15,10,"I wrote a little bit of JS to help me test some configurations - although I wish I could do more testing, this one did the best out of all my trials"
1,1,3,4,10,20,20,20,20,1,Tried to predict some common strategies and tried to give myself the best odds to win the most matchups.
1,1,3,3,4,10,25,30,15,4,"Attempt at guaranteed victory at higher than average but not too high values in hopes opponents would go all in on high value, and I could get more blue chippers for a higher total number. Put low numbers on other targets just in case opponents went even lower for some ""luck"" victories."
1,1,3,3,3,9,25,25,30,0,Punting 10 figuring everyone goes after it. Went after a bit of everything else. 
1,1,2,12,21,12,21,26,2,2,"Preventing potential major strategies (e.g. Capture top 4 numbers with 25 each, match expected value of soliders to castles.)  Also 2 soldiers for 10 and 9 attempts to steal these castles from people who dont make an attempt on them or try to steal with only 1.  Covering numbers like 8,7 and 5 because they are in a majority of sets that would lead to a win.  Appologies for brevity and typos, on a cell phone on a plane hurdeling down the runaway."
1,1,2,12,5,21,2,24,2,30,Some guesswork and pseudo-statistical modelling on how to reach 28...
1,1,2,11,5,7,19,11,22,21,"I randomly assigned troop deployment values for 100 warlords, simulated the choose 4950 battles, and then chose the deployment strategy that won the most wars."
1,1,2,11,5,7,9,11,22,21,"I randomly assigned troop deployment for 100 warlords, simulated the 4950 wars, and then chose the deployment that won the most of their 100 match ups. Note, I am re-entering my answer out of concern that I miss-typed my original answer (it may have added up to 110)."
1,1,2,10,12,15,25,30,2,2,"Winning the game requires 28 points.  This means that even winning castles 8, 9 and 10 is not enough to guarantee victory.  I want to maximize my potential avenues to win by concentrating on castles 4 through 8.  If I win all three, I have 30 points and guaranteed to win.  The biggest weakness is giving my opponent an easy path to 9 and 10, but by spreading out my attack I have more options.  "
1,1,2,10,1,15,12,19,21,18,This is the top random dog after very many iterations. I don't think there's a stable choice. Some randomness has to prevail.
1,1,2,6,12,18,24,30,3,3,I focused troops on castles 4-8 as winning all of those is sufficient to win. I then scattered a few troops elsewhere in case my opponent had not sent any or very few troops to those castles.
1,1,2,5,10,1,20,20,20,20,"Go for an even distribution between the top 4 castles on the assumption that a lot of people will more heavily weight towards the top and decrease gradually as the castle value decreases, skip 6 as it's the last in the 'upper tier' of castles, and scatter some troops around the lower values to try to pick them up."
1,1,2,5,7,9,11,16,21,26,I just winged it
1,1,2,5,7,8,10,14,22,30,Gut
1,1,2,4,14,21,27,27,2,1,"Assuming some will split evenly and others load up high, I am trying to make sure also possible castles that remain unguarded can be one, but focus on higher side below most highly picked choices and those of little value."
1,1,2,4,6,10,16,25,34,1,"Start with Fibonacci numbers which goemetrically increases soldiers per castle. Don't give up any castle without a fight, so at least 1 soldier for each. Reduce Castle 10 to 1 soldier hoping to enphasizing middle-to-higher-numbered castles. Distribute the other 11 soldiers to those castles, so starting with Castle 4, add 1, 1, 2, 3, and 4."
1,1,2,4,6,9,13,17,21,26,Even distribution??
1,1,2,4,6,9,13,17,21,26,"Y=X^2, scaled down so it adds up to 100.  The more a castle is worth, the more attention it receives from other players.  I multiply attention by value to get priority.  I then deploy troops according to priority score.  Trying to win four castles (the shortest path to 28) intuitively seems to be a losing proposition against a large field (I can't guess which four castles everyone will leave least attended), so I will attempt competitiveness at each castle.  I have a healthy amount of gut instinct mixed in with a dollop of math to formulate this approach.

I sure hope I can find out my specific W-T-L record and rank when this is done.  This was a lot of fun!"
1,1,2,4,6,9,13,17,21,26,"Number of troops assigned to each castle is proportional to the square of its point value. One soldier was left over due to rounding, and was assigned to Tower 1 to avoid leaving it completely undefended."
1,1,2,3,5,8,13,21,34,12,Can't go wrong with Fibonacci!!
1,1,2,2,20,20,25,25,2,2,Wanted middle and a shot at the top and bottom
1,1,2,2,16,20,1,26,30,1,"""What is of supreme importance in war is to attack the enemy's strategy"" -Sun Tzu: The Art of War

I chose to focus on trying to get outright victories in 5, 6, 8 and 9, since winning those Castles would give 28 points, assuring me of head to head victory no matter what happens with the other castles. I also took a small portion of my troops (8 troops) and allocated them semi-randomly to the other castles, in an attempt to set myself apart from (and hopefully above) anyone who would try a similar 5,6,8, and 9 strategy variant.
"
1,1,2,2,4,6,10,14,23,37,"There is no equilibrium best strategy.  (Suppose such a strategy existed, given by s(i) for i in {1,...,10}.  There is some i for which s(i) is at least 9, call it n.  Let t(i) = s(i)+1, except for t(n) = s(n)-9.  Strategy t beats strategy s, 55-n points to n points.)  If we were running a tournament I would look into some sort of tit-for-tat based solution that could try to take advantage of the meta-environment by basing each round's strategy on the enemy strategy of the previous round--though even that would be insufficient, since if you can guess your opponent's strategy you can always defeat it.  With the intended structure being to run each strategy against each other in isolation, even that isn't available for use.

The strategy I have submitted is 2/3 of the Fibonacci numbers, rounded up, with the last troop going to castle 7 (to beat the even split strategy)."
1,1,2,2,3,15,20,25,30,1,"I figure too many people will go for 10, so I won't waste anything other than a token troop on that one. You need 28 points to win a round, so I loaded up on 6-9 to try to take those."
1,1,2,2,2,11,21,24,34,2,"I expect most people to use some kind of progressive system, with more soldiers in higher-value castles, so I was aiming for an approach that would work against almost all 'rational' approaches that involve a flat distribution or a monotonic increase with castle value. My system places soldiers in all castles to win or tie against undefended or poorly defended castles. What I really hope serves me well is to cede castle 10 to anyone who doesn't heavily underweight it, and place the large majority of my troops progressively in castles 6-9, with endpoints based on 'round' numbers plus one (10+1 in #6 and 33+1 in #9). I hope to usually win at least 3 of the battles for castles 6-9, and pick up the additional 4-6 points to get to 28 from the other 6 castles, if necessary."
1,1,2,2,2,2,30,30,30,0,
1,1,2,2,2,2,29,29,29,3,"Evolutionary algorithm. I made 15 sets, and then kept replacing the worst performing one with one that would beat the best one."
1,1,1,30,1,1,1,62,1,1,"(1+2+3...10) = 55, therefore need to win 28 value of castles to win overall.

Minimal number of castle is 4 (10,9,8,1) or (9,8,7,4) so assuming many people will pick one of those it might be possible to pick up 2,3,5,6 for 16 points with only 1 point per castle. 

This leaves 12 points left, which make for 4 & 8 required. People probably weight towards the high values, so splitting them 2:1 the same as the points weighting seems sensible. 

After further consideration of the split for ties decided to reduce each primary force by 1 to take full possession of any castle ignored by my opponent. Giving me 8 x 1 and one 62 and one 30 and to hope that I've guess what other people will try correctly. 

"
1,1,1,23,1,1,23,24,24,1,This was the minimum number of points needed to ensure victory after discounting Castle 10 as I figured people were most likely to try and capture that one. I included 1 soldier per castle just to pick up any free points in case someone left it unguarded or attempted my same strategy.
1,1,1,20,1,1,24,25,25,1,"With ten castles for grabs worth one to ten points each, I found the total number of available points is 55 and therefore the win condition is amassing 28 points. My first Approach was to find the smalles number of victories that would meet this condition, which is four castles, as even taking castles eight, nine, and ten give only 27 points. Thinking most players would aggressively secure castle ten, I decided focus on the next three most valuable targets and the least valuable castle to supplement it for a win. Initially, I split my pool of 100 into four groups of 25 troops each for castles four, seven, eight, and nine, but decided to reduce the count at castle four by five troops and castle seven by one troop, sending one troop to each of the remaining six castles in case my opponent's plan had similar roots to mine and used dense placement at a few castles and sparse to no placement elsewhere. "
1,1,1,19,19,19,19,19,1,1,"You need 28 points to win. I chose the series of five that gives it to you (4 through 8 gives you 30) and also includes lower value castles (4 and 5) where troops may not be deployed. I equally split the armies. Then, as a reserve in case of a warlord punting any of the castles I'm not really attacking, I put 1 troop to secure the points. "
1,1,1,19,1,1,25,25,25,1,"put (almost) all the eggs in one winning combination, but don't give away any freebies"
1,1,1,17,18,19,20,21,1,1,"The goal here is to claim 30 points, and give up the remaining 25.  I feel like many people will try to claim the 10 and 9 castles (with them being the most valuable targets), so I'm focusing on castles 4-8.  I did place one soldier on a suicide mission to each of the remaining castles, just in case someone decided to completely abandon a castle I can claim a free VP or two."
1,1,1,16,1,1,26,26,26,1,"This essentially an electoral college problem - a race to 28 points.

Mostly ignore 10 (California) so opponent's expected high deployment there is an inefficient use of resources. However, use 1 in every castle in case it is ignored.

I choose 9, 8, 7 and 4 as my path to 28.

I also went with 26 instead of 25 in 9, 8, and 7 to counter someone trying to do the same strategy with an even split of 25 in each castle. 4 gets less troops because its not really a critical path to winning, especially if I pick up some other small ones which were ignored."
1,1,1,15,20,20,20,20,1,1,"Sacrifice the top 2 to give a better chance at holding the next 5 highest which would be enough to guarantee victory. Also, put at least 1 in every Castle in case opponent doesn't bother defending."
1,1,1,15,20,20,20,20,1,1,"Reasoning was psychological, with expected opponent preference for either near-exclusive focus on valuable castles or an even spread. This distribution is designed to defeat both approaches. At least 1 point in each punishes exclusive focus anywhere, while grabbing all mid-value castles will beat the top two, or three if you get all the low-value ones as well."
1,1,1,15,18,1,27,1,34,1,Put just over (castle VP number)/(28 points needed to win) at four locations totaling 25 VP with expectations of getting the remaining 3 VP from tying or winning another castle where the opponent has not placed troops
1,1,1,15,17,19,21,23,1,1,"55 possible points to be won, figure winning 30 points gets a win against most opponents. Let the opponent have the high point castles (9+10) to try and guarantee 30 points from the middle castles. At least 1 soldier at each castle in case the opponent has any zeroes"
1,1,1,15,15,15,16,16,10,10,Grab the middle and hope for the best
1,1,1,15,1,1,26,25,28,1,"I only need 28 points to win. Thus, I can spread my troops out over 4 castles (10, 9, 8, 1 or 9, 8, 7, 4). I chose the second option as I believe I am more likely to lose the 10 castles. Secondly, I want to win all the vacant castles by the other player. Thus, I can win even if I fail to secure all the castles I plan. My distribution was mainly based on a pro rata points spread with the exception, that if an opposing player distributed his/her castles pro rata over castles 1-7, I would still secure both and win."
1,1,1,14,16,19,22,24,1,1,Essentially abandoning the high point castles allows for many troops in castles required for majority.
1,1,1,14,15,16,25,25,1,1,"I tried pick a strategy that could counter strategies such as uniform deployment and top end (7,8,9,10) deployment. I focused on the middle castles, but left at least one troop everywhere to get free castles in case others leave them open."
1,1,1,14,14,14,14,14,14,13,"maximize chance to win 7, concede 3"
1,1,1,14,1,1,24,27,30,0,Voodoo
1,1,1,14,1,1,18,26,36,1,Safety Case
1,1,1,13,16,19,22,25,1,1,"Basically, give up 1,2,3,9, and 10 to anyone willing to fight over them (but not for free - punishing anyone who focuses too hard on specific castles) while committing more to the centre castles 4 to 8. If 4 to 8 are won, the majority of the points are won."
1,1,1,13,16,18,22,0,28,0,"I wanted to only go for 28 of the 55 points, but I hedged my bets by bidding 1 for the low values, hoping that I might win easy points."
1,1,1,12,17,18,20,26,2,2,"Tried to decisively win in the 4,5,6,7,8 range (if all wins, will definitely win), while devoting minimal resources to 1,2,3 and 9/10 but enough to take them if minimal support from other team. "
1,1,1,12,15,19,23,26,1,1,Duck the 9 and 10 but try and dominate enough to get victory.
1,1,1,12,14,16,17,18,19,1,"Sacrifice 10 (most likely contested) and low scoring castles, with token soldier in case someone did what I do but leaves 0.  Stack all middle tier castles with increasing quantities slightly proportionate to benefit."
1,1,1,12,12,9,1,1,31,31,"It was not nearly as systematic as I would have liked to have time for. But I designed a similar adjudication tool as you describe you'd use in excel. Started with testing a strategy of comparing single troop deviations from an initial point-value weighting. This led to the realization that I'd have too many iterations to test this searching for the local maximum to fit in m excel.

Knew that you just needed a winning coalition, not to represent on each castle (left one soldier on each to cut from cheaters, with slightly similar strategies, though thinking I maybe should've put two in each). Focused bulk of troops on 9 & 10 (my plan pretty much needs those to win, and realized I'd just need to pick up two of 4, 5, or 6, so put some troops there. 

I'm sure those who had more time probably tested their solutions against this type, but it's what I got."
1,1,1,12,1,1,20,25,37,1,Focus on getting 28 points and sacrificing the 10 point castle.
1,1,1,11,14,18,21,1,31,1,"I didn't want to focus on the big point value castles because people are going to waste a lot of soldiers on those unnecessarily. I went for the middle range ones, with a decent amount of soldiers on the 9 castle so that I could win one big one. I put 1's in 1, 2, 3, 8, and 10 just in case someone is foolish enough to not send any soldiers to a castle. Against both a uniform strategy (10 on all castles) and a weighted average strategy, (weight each castle according to its point value, then send a proportional amount of soldiers to that castle), my strategy wins 31 to 24. "
1,1,1,11,1,1,27,28,28,1,Picked a simple route to 28 and defended it without leaving any uncontested.
1,1,1,11,1,1,21,21,21,21,
1,1,1,10,14,18,24,20,10,1,"I am guessing a lot of people will overvalue castle 10 and to a lesser extent 9, so I am all but seceding those castles. I am hoping to win a majority of the middle ground by placing a disproportionate amount of troops there, and placing 1 troop everywhere else to beat others who are trying the same strategy as I am."
1,1,1,10,10,1,20,25,30,1,I wanted to put at least one soldier on every castle to try to win some without much resource allocation to them or at least split them with anyone else having the same idea. I put more soldiers on castles that I thought would give me the best chance to get to the key 28 points required for victory. Also I love the idea of the community interaction in the riddler this week!
1,1,1,9,11,13,18,20,25,1,
1,1,1,7,15,15,15,15,15,15,"I didnt really have a plan, but thought that I should evenly focus my troops on the higher level ones to get a better chance and put a single troop on the low ones in cas the enemy put 0"
1,1,1,7,15,15,15,15,10,10,I was looking to spread over my unites in as much of the middle numbers as possible.
1,1,1,7,12,14,16,21,27,0,"by sacrificing 10 I am hoping the enemy sends the bulk of their army to that castle and I can outright win the rest of the castles, I did not care about the lower castles but sent 1 to each just in case the enemy also did not care about the lesser castles"
1,1,1,7,11,15,20,21,22,1,"sacrifice #10, but load up on 5-9, knowing that 9 + any 3 of the other 4 will win it for you. also, putting 1 on 1, 2, 3, 10 just in case those were totally neglected by my enemy "
1,1,1,7,10,12,14,16,18,20,"Biased towards the more valuable castles, but not ignoring the high end.  Mostly ignoring the bottom 3, but one troop just in case."
1,1,1,7,10,12,14,16,18,20,"1 more than evenly dividing troops by points would suggest for 6 to 10. Same number as even division for 4 and 5. 1 troop each for 1, 2, 3."
1,1,1,7,8,17,26,37,1,1,Clustered at 4-8 to shoot for 55% of points hoping that most waste troops at 9-10. Put at least one in everything to grab any points from those who put zeros.
1,1,1,6,14,16,18,20,22,1,"I will concede no castles outright.  In placing one soldier on castle 10, I hope my opponent will have a lot of soldiers there -- those become 'wasted' soldiers.  To compensate, I put what I hope to be sufficiently heavy proportions on castles 5-9 to carry a good number of points."
1,1,1,6,10,14,10,19,15,23,Used a program to brute-force the best statistically best possible arrangement of soldiers using a smaller number of total soldiers. Reran the program with a higher number of soldiers and the additional requirement that any arrangement of soldiers had to also beat the previously best considered arrangement with some fuzziness. Extrapolated that second arrangement up to 100 total soldiers and fudged some numbers.
1,1,1,5,11,16,28,29,3,5,Because my buddy Tony showed me his plan and this one beat it. 
1,1,1,5,7,11,15,17,19,23,"Intuition, whim, and some very amateur gamesmanship."
1,1,1,5,5,10,1,25,1,50,It won the most random matchups I saw
1,1,1,4,1,1,25,30,35,1,Linderman said go for it!
1,1,1,3,6,11,18,23,22,14,"Poisson distribution around SQRT(10), with at least one troop per castle"
1,1,1,3,3,19,20,20,31,1,Assumed highest troop concentration would be castle 10 -- want to try and win castles 6-9
1,1,1,3,1,23,23,23,23,1,
1,1,1,2,10,12,16,17,19,21,Minimum of 1 to each to win any low hanging fruits -- then emphasize higher value castles
1,1,1,2,3,6,9,14,24,39,Mimics Fibonacci series. I approximated the percentages for the first ten numbers in the sequence to give higher strengths to higher value castles. 
1,1,1,2,3,5,20,21,22,24,"Try to win 7,8,9,10, and any other unchallenged castles."
1,1,1,2,3,5,10,20,21,36,"Fibonacci sequence with modifications. Aiming for 28 total points. Playing against a few different strategies. In a simple top 5 get 20 strategy, this wins the bottom castles plus 10 for the win. In an even spread, this gets 34 and wins. "
1,1,1,2,2,12,14,17,24,26,"I wanted to earn points if someone didn't put any troops at a castle, so I put minimal troops at the first five. Then I just tiered up the troop level at the rate the points go up. I may not win, but I'm confident I can go above .500"
1,1,1,2,2,11,13,1,23,45,"I worked with a few assumptions- You must get to 28. To do this you have to win outright a minimum of 4 towers. (10,9,8, and 1) It is always better to try for every tower with at least one troop. There is also never going to be a perfect strategy which beats all other strategies.  The best way to win will be by countering the most common strategies. The higher value towers are worth the most, but getting two lower towers will more often than not, net more value- 10 is equal or worth more than 1/2 of the tower value combinations, 9 is worth less than 1/2 of the combined pairings and so on. I based my selection loosely around the Fibonacci sequence (very loosely) and then chose 8 as a tower to essentially pass on so that I could maybe out guess some of the strategies which simply tried to win the higher towers. Here's hoping, To Battle!"
1,1,1,2,2,6,26,26,19,16,Since you need at least 28 points to win the war to win with the minimum number of castles you need to win 4 castles all of which are 6 points or higher. Knowing that distributed troops to beat most deployments that split troops evenly across 4 castles as well as to beat other potentially common distributions. Put some troops in lower value castles to try to reduce likelihood of splitting points in those areas reducing the likelihood of losing one higher point castle leading to losing the war.
1,1,1,2,2,2,2,2,42,45,"The Monarch of Riddler Nation will be the player who wins the most match-ups - only victory and defeat matter, not average scores. A winning strategy will be one that maximizes the number of (often very narrow) victories.

Therefore, one could do very well by concentrating all troops evenly among any combination of four castles (25 soldiers per castle) that together yield 28 or more victory points. There are 27 such combinations, all of which win against anyone who tries to spread troops more thinly among all 10 castles (e.g. [1,1,1,1,1,1,23,23,24,24]). The dominant strategy among these is [0,0,0,0,0,0,25,25,25,25], which defeats any other 4 x 25 combination (e.g. [0,0,0,25,0,25,0,25,0,25]).

I think enough people will start with a variation on 4 x 25 that it is the strategy to beat. The only successful counter involves piling more than 25 soldiers into a single castle, at the expense of potentially conceding another castle. How does one beat [0,0,0,0,0,0,25,25,25,25]? Easy! [0,0,0,0,0,0,24,25,25,26]!
Or better yet, [0,0,0,0,0,0,0,33,33,34]! And how does one beat that? [0,0,0,0,0,0,0,0,0,50,50]! But wait, that could be defeated by [0,0,0,0,0,0,0,0,0,100]!

If we take this to its logical extreme, the equilibrium strategy is to pile all soldiers into the highest value castle. This is because any strategy contesting a certain set of castles can be outdone by another that contests the most valuable of these castles *harder* at the expense of any single less valuable castle.

However, will this actually be successful in reality? I highly doubt it. An iterated series of matches between two opponents might very well result in this strategy being used, if only for a brief moment. Opponents stack troops in the 4 most important castles until one realizes targeting only the top 3 is even better, and so on until only the 10 victory point castle is contested. But this strategy would immediately be handily beaten (45-10) in the next round by an opponent who suddenly decides to send soldiers to all castles again.

Just like in the Keynesian Beauty Contest problem, most players will not take this game to its logical extreme. Here, anyone who contests only castle 10 will probably lose the vast majority of their contests. They will only beat those who contest a set of other castles with a combined value less than 10, which I think will be rather rare.

I think it's quite likely that most contestants will invest heavily in 3 or 4 castles. Stacking an absurd amount of soldiers in castles 9 and 10 can defeat most such strategies by snatching 19 victory points right off the bat, with hopefully enough VP coming to me from ties or narrow victories at lower VP castles.

This leaves me very vulnerable to very basic strategies (e.g. 10 soldiers per castle, or random placement) or to those that stack a minimum of 3 or 4 soldiers per castle, but I am hoping the presumed preponderance of 4 x 25 type strategies will scare people off from these types of even spreads.
"
1,1,1,1,31,26,1,21,16,1,"Saw this idea used in ""Kelly's Heroes""."
1,1,1,1,30,35,30,1,0,0,Luck.
1,1,1,1,24,24,1,23,23,1,I can't spend anymore time on this stupid castle game. I need to get back to work. This is the count I had in my Excel sheet when I decided I'd spent too much time on this.
1,1,1,1,23,23,1,24,24,1,Decided to play electoral college with this and focus on the 4 numbers that would get me half. I put 1 everywhere else to thwart other people doing the same thing 
1,1,1,1,23,23,1,24,24,1,"I didn't have any great ideas. But you need 28 points to win - might as well go for exactly 28. I think a lot of people will load up on 10. After that, just guesswork. Putting at least 1 on each castle is a cheap investment. "
1,1,1,1,23,23,1,23,24,1,"The goal is to get get 1 point more than half the points available (28). Therefore, it is mostly a waste of resources to allocate troops to getting additional points, an advantage to concentrate resources only on the castles you need to win.

The minimum number of forts needed to get this point total is 4.  I need to guess which set of 4 castles is the least likely for the majority of players to concentrate their forces on.  Castle 10 is an obvious ""honeypot"", so I'll avoid concentrating there. Castle 7 is the overlap between the needed ""high castles"" (7-10) and ""low castles"" (1-7). A good combination is then perhaps 9, 8, 6, and 5 (total 28).  I will mostly concede all the other castles, but since some people may totally concede a castle if they are concentrating forces as well, I will leave at least 1 troop at every castle.  Since I can't divide 93 evenly between 4 castles, I will give an additional troop at castle 9 since it's most valuable."
1,1,1,1,21,21,1,26,26,1,"winning 5, 6, 8, 9 wins"
1,1,1,1,20,21,1,26,27,1,"Punted on the 10, concentrated troops to get 28 points (minimum to win), deployed 1 troop to remaining castles just in case."
1,1,1,1,20,21,1,26,27,1,Fill all with at least one - hopefully easy points - and take an unconventional path to 28 that didn't use the 10. 
1,1,1,1,20,20,20,1,1,34,"So, there are 55 points, meaning that to win, one must get to 28.  1-7 seems too obvious, so I reject that.  I picked 5,6,7,10 because it seemed like a nice set.  I put 1 in each of the others to pick up some cheap points against people who zero them. Really, though, I'm just guessing.

I think that a more interesting problem would be to design an allocation strategy for the case where one sends troops to Castle 1, sees what the opponent did at Castle 1, and then moves on to Castle 2. That is, the strategy could adapt as the battle continues. "
1,1,1,1,19,19,19,19,19,1,"Avoid castle 10 as many will put a lot there, and divide most among the next 5 castles that are more than enough to win.  Leave 1 each in the remaining castles (including 10) to win points against those that leave 0 in some of castles, or also leave 1 in some of those castles."
1,1,1,1,18,19,1,26,31,1,"28 points are needed to ensure victory, which requires at least 4 castles. The optimum way to get there is 9, 8, 6, 5, so my main goal is to win those. I tried to account for players using strategies that focus on higher castles by weighing 9 and 8 more. I also threw 1 soldier on all the other castles - it decreases my forces to use on the four castles I want, but it may give me big gains against anyone using all of their soldiers on 4 or 5 castles. "
1,1,1,1,17,20,1,27,30,1,"55 points in total.
28 points to win.
pick 4 (least castles need) to achieve 28 points.
Avoid going for 10 because probably many people go for 10.
leave 1 on rest of castles in case people put down 0 troops there.
roughly leave rest of troops (94) proportionally on those particular 4 castles"
1,1,1,1,17,18,19,20,21,1,"Put 1 troop where I suspected many may put 0 (and thus win those castles with limited investment), and then spread the rest out relatively evenly but with the slightest skew of troops towards the higher values."
1,1,1,1,17,17,2,21,37,2,"This is a revision to my initial strategy. 9,8,6,and 5 are really what Im shooting to win. This also accounts for anyone that may disregard castles 1-4 and evenly distribute soldiers in 5-10."
1,1,1,1,17,17,1,26,34,1,"Primary strategy is to win 4 numbers to get to 28. I chose 9,8,6,5. Psychologically I assume people will go crazy to win 10 so I avoided 10. I set my armies to beat almost all simple strategies using other sets of 4 numbers (4 25's or 3 33's + 1 or even split from 1-7). I lose to 25 armies on 10, 7, 6 ,5 but you can't win them all. The spread out singles are key to beating people that leave castles undefended, allowing me to lose 9,8,6,or5 and still win. Analyzing strategies might be a fun topic for a follow up column . . . "
1,1,1,1,16,25,2,26,26,1,
1,1,1,1,16,24,25,28,1,2,"Firstly, I need to make sure to send 1 to each castle to pick up any freebies. 9 and 10 are the obvious castles, so I'm hoping my opponent overcommits. I spread the rest around the remaining high points in descending order because I suspect that is what my opponents will do, although hopefully with fewer remaining soldiers after those allocated to 9 and 10."
1,1,1,1,16,20,1,27,30,1,"Put 1 soldier in each castle in case an opponent doesn't put any, to get an ""easy victory.""
Pick castles representing a bare majority of points, and focus all remaining resources there.  Don't pick 10, or a string of numbers, as those seem too obvious.  Focus forces proportional to value.
"
1,1,1,1,16,19,1,28,31,1,"Victory requires 28 points, which requires a minimum of 4 castles. The sum of any castle and the 3 above it exceeds 28 points beginning at castle 6 (6,7,8,9), and then by 2 points. High value castles will be more competitive than low, so dropping castle 7 in favor of 5 places as many castles as possible as far down the list as possible. Weighting troop commitment by point value yields 18,21,29,32 for castles 5,6,8,9. In this case all castles are must win, one victory condition. Risking castles by diverting 6 troops (2 each from 5,6 and 1 from 8,9) picks up any remaining castle where no troops were committed by the opponent, laying claim to any of the remaining 27 points that might compensate for a tie or loss in the big 4. If the enemy also sends a token troop to every castle I still gain 13.5 points, within half a point of compensating for the total loss of any two of my must-win castles except for both 8 and 9. The solution is not rigorously tested, but it provides ubiquitous coverage while focusing maximum troop strength on the easiest targets necessary."
1,1,1,1,15,19,23,1,1,37,"If you want to win, you must acquire 28 or more points. This requires that you should spend approximately 3.57 units per point that you wish to acquire. Regardless of choice, each castle should have at least 1 unit assigned so to capture castles that the other player does not find interesting for their win strategy. 

My choices were based on the competition for 10, 9 and 8. I found that 10 is probably the most cost effective castle to win, but will also be heavily contested. Most would allocate either way too many points to this castle and leave others unguarded, or would allocate less than the optimal rate (rounded up) + 1. 9 and 8 are also essential to victory for a minimum castle set (4 castles, adding up to 28 or more), but without 10, require 2 more castles. I decided to avoid 8 and 9 based on the fact that if they wanted 10, they might also want 8 and 9. So instead, I focused my troops on  5, 6 and 7 to achieve my minimum of 28. 

I assumed that other players would either ignore the castles they did not want, in favor of the castles they did want, or they would allocate a minimum of 1 troop as well, in which there would be no point advantage to the castles they did not want. If only 1 troop is allocated to castles with little value to the strategy, then you have a maximum amount of troops to allocate to the 4 castles that you intend to take. "
1,1,1,1,15,19,1,26,34,1,"I want to win 28 points as comfortably as possible, and it can't be done with 3 castles. It can be done many ways with 4 castles, but 9+8+6+5 is the way that avoids the critical Castle 10; avoiding Castle 7 (versus playing 9+8+7+4) seems like a better choice as well. By playing 1 in each castle, this will also pick up points against other overloaded strategies that are likely to send 0. Castle 9 is set to be larger than 1/3 to beat heavily overloaded strategies, though it might be better to play it completely proportionally (17, 20, 27, 30 instead of 15, 19, 26, 34)

This strategy will soundly beat ""balanced"" strategies or strategies that overload Castle 10. It's still beatable for sure, but I think it will do well.

Thanks! This is a great idea."
1,1,1,1,13,13,1,25,41,3,"If I can take 9, 8,6, and 5 I win. However, I dont want to leave the others ungaurded."
1,1,1,1,12,18,20,22,24,0,"Not worth fighting for 10 as I assume needed troops for this will be high.  Hoping to win Castles 8 & 9 as well as two from 5, 6, & 7.  Placed just 1 in the lower Castles in case of (practically) free wins."
1,1,1,1,12,15,25,40,2,2,Need 28 points to win; expecting overwhelming force directed against highest castles.
1,1,1,1,11,21,21,21,21,1,"Casle 1, 2 and 3 are not that important. A lot of people will try to throw some casles. People like round numbers. A lot of people will try to win casle 10."
1,1,1,1,11,13,15,17,19,21,"If Castle # is N, I wanted to beat everyone who 2*N'ed the top half of the castles.  Doing so left me four four extra soldiers, and I put them down in the four bottom castles as maybe there's some easy pickin's there."
1,1,1,1,11,13,15,17,19,21,"I figured a stat-head would find a formula for best-case distribution of troops. I decided to up my soldiers in the higher places above what I thought was the statistically smart ""average."" I tossed one in each of the lower castles to simple snipe some points from anyone who leaves them empty, thinking them not worth the soldiers.

In simple terms: I'm guessing."
1,1,1,1,11,13,15,17,19,21,"The goal is to get to 28 points, but because there will be so many different strategies, my goal is to get as many points as possible. "
1,1,1,1,11,13,15,17,19,21,
1,1,1,1,11,12,21,21,31,0,Over-reliance on anticipation of the opponent's maneuvers. 
1,1,1,1,11,11,11,16,21,26,"One more than common round numbers on each with more focus on higher numbers. I used a loose model to estimate the expected value of several strategies against each other, and this was most successful (but it could just be effective against straw men)."
1,1,1,1,11,11,11,11,51,1,"I tried to come up with a strategy that took into account a few different principles:

(1) Win in at any unchallenged castle (i.e. at least 1 soldier to every castle... the pawns muahaha). 

(2) Counteract what I believe are high probability strategies (like at least 1 on every one [ha!], 10 evenly distributed throughout all, or heavy fights for 9 and 10)

As a law student, my strengths aren't math, but I did my best to figure the initial likelihoods of soldier distribution and adjust accordingly to evidence (just for example, I assume initially I'm going to lose the 10 castle 99% of the time, .9% of time I'll split it with someone, and .1% of time I'll win it; there's some hope that your saavy readers do the math and maybe put only 1 in that category b/c they assume like I do it's a lost cause but that they can't just toss it, putting 1, which increases probability that 8 or 9 is a battle ground). Tried to consider which categories I could lose and still come out on top with if I won my expected, and what I'd have to win if lost one of my expected. With more time I might actually do some kind of more consistent  Bayesian calculation but alas, once more into the breach!"
1,1,1,1,11,11,1,31,41,1,"In order to win this game,  you need to get the majority of points. 1+2+3+...+10=55. 55/2= 27.5. So the least amount of points you need to win is 28.

There are lots of ways to getting to 28.1+2+3+...+6=21 so you need at least castle 7 to win. This also means you are guaranteed to win with 7 castles. 10+9+8=27, so you need to win at least 4 castles as well. If you are only going to use 4 castles, you need castle 9 and/or castle 10 because 8+7+6+5=26.

My first strategy was to have at least one soldier go to each castle. That way if my opponent decides to only attack 4 castles, I am guaranteed 6.

My second strategy was to try to win with only 4 castles.  Since you need either castle 9 or 10, I figured they would be the most desired and soldiers would be sent accordingly. So instead of guarding both,  I decided to pick one. I decided to go with castle 9 and not 10. After that,  my options if I wanted only 4 castles were either castles 4, 7, 8, and 9 or castles 5, 6, 8, and 9. I went with the latter so I wouldn't have to win 7, 8, and 9.

Once I chose my 4 castles, I had 90 soldiers left to play with and sent them by weight.  40 more to castle 9, 30 more to castle 8, and 10 more to both castles 5 and 6. I put 10 more in those csdtles, in case someone just put 10 in each castle."
1,1,1,1,10,20,20,20,15,11,"11 on 10 to beat anyone who went 10 across the board. Less on 9 than 8 incase someone heavily tries to win the last two - can't compete - need to spread a bit. 6,7,8 with 20 is a decently powerful base to take which is likely to get at least one or two of them. 10 on 5 as it's worth something, then 1 to all the others - I mean it's at least worth trying - they total 10 together so I'd rather stock up on the higher ones."
1,1,1,1,10,10,20,20,35,1,
1,1,1,1,9,15,1,25,45,1,"I picked what I believe to be the easiest way to acquire 28 points, the amount required to win the battle. One needs to capture a minimum of 4 castles to achieve this, so this caused me to focus my troops on the four castles that would in my mind be most effective in achieving this. 

By giving up castle 10, it does not make any sense to pursue castles 1-4: their values are equal and it would be a waste of troops to capture one only for the other to offset the gain. To me, the only castles that matter are 5-9. Thus, I deployed troops in the way to most likely take the higher value targets while also hopefully overpowering the lower priority castles with troops gained by sending the bare minimum to half of the castles.

Also, the marginal value of 1 troop is so low that sending one to a castle on the chance that no other troops are sent seems to make sense."
1,1,1,1,7,22,22,22,22,1,"Abandon 10 because most people will invest too heavily there, send at least one to each place to give a chance to gain the areas with the highest likelihood of intentionally abandoned points, spread rest of soldiers equally across highest point locations 6 7 8 9, having at least 22 at each  (which attempts to one-up the approach of adding one to the seemingly round number 20)"
1,1,1,1,6,15,20,20,34,1,"Came upon this deployment with testing in spreadsheets. High value placed on high value castles while ensuring that there's at least a soldier at every castle to avoid splitting points with strategies that completely ignore lower value castles.

Came upon this strategy while testing out scenarios in sheets. I figure the 10 castle itself will be overvalued while the bulk of points come from 6-9 themselves. If my opponent overcommits too much to the 10 castle, I will easily sweep the rest of the castle points.

I always send at least one soldier to ensure I don't split points with any castles my opponent abandons."
1,1,1,1,6,11,16,26,36,1,"I abandoned castle 10 in hopes of saving troops where the other team might send a larger force. Focused more troops on castles 6-9, since those are more valuable, but saved at least 1 troop for the low value castles and castle 10 in case the other team also sent no one there."
1,1,1,1,6,11,16,21,21,21,"Send at least 1 solider per castle;  use ""1/6"" ending numbers on the assumption that many players will round to 0/5; send majority of armies to the highest-valued castles."
1,1,1,1,6,11,12,33,34,0,"Think of this as bidding at a silent auction instead of troop deployment. The value of Castle 10 is high enough that there will be lots of bids - leaving me a very small chance of winning it. So I put my resources elsewhere. The idea in general is to invest just enough to win each castle except Castle 10, but no more than necessary."
1,1,1,1,3,30,30,30,1,1,"People will prioritize castle 9 and 10 to get 19 points, but if I win 8, 7, and 6, I will get 21 points."
1,1,1,1,3,30,30,30,0,3,Gut feeling 
1,1,1,1,2,4,6,12,24,48,
1,1,1,1,2,2,7,11,23,51,"A game can be modeled in normal form. After solving the toy problems of (2,1), (2,2), (3,2), (2,3) where (troops, castles). One only wins in the upper triangle of the matrix. Using iterated domination can only reduce the real game space a bit. If one assumes there is a positive probability of the opponent playing all of the non-strictly dominated strategies, then a best-response is to mix over the available strategies with probability of winning as the mixing probabilities. I used a RNG, random module in python, to draw a deployment of troops in the set of feasible solutions. Then transitioned using pairwise castle changes until it was satisfactorily burned in ensuring that a positive number of troops were allocated to every castle. "
1,1,1,1,1,31,32,32,0,0,I'm trying to take advantage on the bottom 5 of people leaving those as 0. I'm also trying to beat people who put high amounts in 9 and 10. 8 7 6 are worth more combined than 9 and 10
1,1,1,1,1,31,1,31,1,31,Randomly chosen 
1,1,1,1,1,23,23,24,24,1,"Assume opponent uses resources on castle 10, but needs 18 more points to win, but can't get there without 6,7,8,9 which are heavily fortified."
1,1,1,1,1,21,36,36,1,1,"9 and 10 are too obvious, and people will deploy way too many to take them. 6, 7, 8 are more likely higher yield targets, 1-5 are worthless. I put one in them, just in case someone put 0."
1,1,1,1,1,21,21,26,26,1,"Loading up on 6 through 9 guarantees a win if all 4 of these numbers are won. Conceding the 10 to top-heavy strategies improves my chances of winning 6 through 9.Leaving 1 on the remaining spaces allows some versatility in the event that the opposing strategy involves loading up on some of the same numbers. This strategy can beat many strategies that I expect to be common, including ramps and top-heavy distributions."
1,1,1,1,1,20,24,25,25,1,
1,1,1,1,1,20,22,24,28,1,"I think most opponents will use some sort of strategy of going for the most valuable castles or mathematically increasing the number of troops in each castle. With this strategy I hope to beat both opponents in castles 6, 7, 8 and 9 and win the war that way, while sacrificing castle #10 but at least having a presence everywhere in case anyone disregards a castle completely."
1,1,1,1,1,20,20,27,27,1,"If I win 6,7,8,9 I will win everytime. I'm planning on people targeting 10 too much."
1,1,1,1,1,19,19,19,19,19,"I want to win the top 5.  If they put everything in 10 and 9;  6,7,8, will still beat them.  If they go 10 even, 6-10 will beat 1-5.  I am hoping this is a good guess!  :)"
1,1,1,1,1,19,19,19,19,19,"My college Game Theory professor has his students play this game (in a variation where each castle is 1 point.) He also keeps a database of students' previously used strategies.

Using this knowledge, I modified the code for this scoring system and created a bunch of copies of a couple strategies I expect people to deploy for this contest. This strategy held up pretty well against many variations that I added to the database."
1,1,1,1,1,17,18,19,20,21,"Try to 1 up the opponent at the top castles. Send a person to each of the bottom so I don't give away ""freebies"""
1,1,1,1,1,17,18,19,20,21,"I'm hoping for opponents with a 'balanced' composition to spend too many soldiers in the lower half and lose most of the upper half to me. If someone did decide to leave some of the lower half unstaffed, I'll snag the free castles without seriously diminishing my chances, and I expect to win against a lot of really top-heavy compositions by virtue of taking castles 6,7,8. To be perfectly honest, I didn't think about this very seriously, which might actually work to my advantage - after all, I'm just trying to beat everyone who thought one step ahead by thinking ahead by two."
1,1,1,1,1,16,26,26,26,1,"Goal is to try to achieve 28 points; Must take at least 4 castles, and at least one of the castles taken must be >6. I expect troop numbers to be frequently multiples of 5. I am willing to sacrifice the biggest castle if it means I am more likely to win potentially easier battles worth almost the same number of points."
1,1,1,1,1,16,21,26,31,1,"I didn't want to fight for castle 10 or castles worth relatively few points.  I put 1 troop on each in case they were completely uncontested though.  I put most of my forces into the high value castles 6-9, weighted more heavily toward the more valuable castles.  In case others used round numbers, I went with quantities ending in 1 and 6 instead of 0 and 5.  I'd love to know how I do, though I don't expect to win."
1,1,1,1,1,16,1,26,26,26,"I'm a math teacher at a high school, we did this as an experiment all day and this deployment ended up being the winner."
1,1,1,1,1,15,20,29,30,1,"There are 55 total points so I need 28 to win.  9+8+7 gets me 24 of them.  I'm punting on castle 10 assuming others will put a lot of people there.  I'm hoping to get three of 9,8,7,6 plus enough other castles to get 28."
1,1,1,1,1,15,20,20,20,20,Just prioritizing the most points but still putting troops on other castles in case the enemy forgoes them
1,1,1,1,1,15,20,20,20,20,"Trying to win 4 of the top 5 as main path to success. If that doesn't work, picking up easy low points against people who tried similar strategies may work."
1,1,1,1,1,15,15,15,20,30,I was willing to sacrifice lower value castles to hopefully win larger castles.
1,1,1,1,1,12,20,20,21,22,"Based on over 1,000,000,000 simulated battles with various different deployments (some random, some seeded), as well as using a simplistic machine learning algorithm to adjust the deployments, this was the most successful deployment, with slight modifications as a nod towards considering likely human strategies.  20/20/20/20/20 in 6/7/8/9/10 was the most successful seed design (beating out 16/16/17/17/17/17, initially the most successful seed, in later rounds as the deployments improved).   I remove some from 6, allocating a few extra to the high end and one to each of the low five, so to beat strategies that place zero on many castles.

General design: started with 10,000 randomly allocated rows allocating soldiers in groups of 5 (to get improved clustering), plus ten seed rows, including 100 in castle 10, 50/50, 10/30/30/30, 25/25/25/25, 20/20/20/20/20, and 16/16/17/17/17/17 (castle 10 always the last number) and several others.  Simulated all deployments battling all others, then tallied wins, and created a new sample of 20k rows using PPS with replacement sampling based on the square of wins.  Reran the simulated battles, then did another sample selection step with 25k rows again using PPS, and reran those simulated battles.  The winner was 19/19/20/21/21 which won 23,812 battles out of 25,000.

I also confirmed the results using a OLS regression with the ten castle quantities as independent variables and wins as the dependent variable.  OLS indicates that castles 7 through 10 have positive coefficients, 6 has approximately 0 coefficient, and 1-5 have negative coefficients, suggesting that more tests might pull towards a higher number for castle 9 and 10.  Given the slightly negative (but nearly zero) weight on 6, I felt confident removing some of the armies from it in order to put single armies on the other five."
1,1,1,1,1,12,16,33,33,1,Instinctively most people will try to defend the 10th castle with the most troops.  This strategy defends the middle castles 6 though 9 with a slightly uneven distribution of troops. Also throwing in 1 at each of the other castles in case the other side simply puts no troops there.  The value of those 5 troopers in Castles 6 and 7 is only slightly problematic.
1,1,1,1,1,12,16,18,22,27,"Probability weighted the top 5 castles, 1 soldier to the others to catch open slots. "
1,1,1,1,1,11,21,31,21,11,Why? Because you asked me to.
1,1,1,1,1,11,21,21,21,21,Balanced attack of top castles.  Defeats the uniform and weighted by points distributions. 
1,1,1,1,1,11,15,34,34,1,Best strategies seem to be trying to get castles 6-9 or 8-10 and 1 of anything else. This strategy does well against both strategies that use troop deployments that seem logically intuitive. Any strategy that beats this one seems to do poorly against what I would consider normal strategies.
1,1,1,1,1,10,20,30,35,0,"Assume that rational opponents will over-commit troops to castle 10, leaving castles 9, 8, 7, and 6 relatively undefended; note that these four castles sum to 30, which is enough to win most battles.  Castles 1-5 are each allocated one troop, to hedge against a similar strategy that assigns zero troops to each."
1,1,1,1,1,7,14,19,25,30,
1,1,1,1,1,7,14,16,20,38,"Not really sure. I want to win the higher ones, but not give up on the cheap lower ones."
1,1,1,1,1,7,7,21,29,31,"Maximixing potential to win the most amount of points on the high castles, while catching people off guard who sent 0 to the lower castles"
1,1,1,1,1,5,15,20,25,30,Pick up as many castles as possible
1,1,1,1,1,1,31,31,31,1,"Always have at least 1 at each castle to claim points in case the enemy does not show up. Go strong for the 9-8-7 castles; people tend to fixate on the 10, so let them have it.  The goal is 28 points.

This is not all that different from the electoral college, only more arrows, battering rams and boiling oil."
1,1,1,1,1,1,26,22,23,23,"Take any uncontested castles, focus on the high-value ones but prioritize 7 since it + 1-6 is enough to win."
1,1,1,1,1,1,23,23,24,24,"It is important to note that you MUST capture at least 1 of castles [7,8,9, or 10] in order to have more than half the points (28/55). To that end, I have sent 1 troop to each of the lesser castles in order to pick up ""free"" points from the people who focus solely on the larger point castles. I have split the remaining troops evenly among the remaining high value castles. I would hope that this means that I dominate similar strategies. I have no doubt that my strategy is not optimal, and I expect to be beaten, but I figured I'd throw my hat into the ring."
1,1,1,1,1,1,23,23,24,24,"Their are 55 possible points so one must win 28 to win. I will send 1 to each in case one person does not send any to a base. Then I will divide the remaining 90 among the 7,8,9,10 which is all that is nescecary to reach 28. 90/4 is 22 with 2 remainder. This will be put on the 9 and 10 to go after the most important targets"
1,1,1,1,1,1,23,23,24,24,"There are 55 points possible, and all that is needed for victory is 28. So I invest heavily in the four most valuable castles, but I also send one soldier to each of the remaining castles so that I'm not beaten by a strategy that grabs one ""core"" castle and picks up a few points from smaller-point castles."
1,1,1,1,1,1,23,23,24,24,If I can win the 4 highest castles it doesn't matter if I lose the other 6.  The 1 soldier in each of the first 6 castles is to win against those who put 0.
1,1,1,1,1,1,21,26,26,21,"I tried to come up with as many possible strategies as I could, then guard against all of them. Some of the strategies that I am hoping to guard against: 1. A relatively even troop deployment: I would hope to win 10, 9, 8, and 7 guarantee a victory. 2. Top-heavy deployment: They would win 10 and 7, but might lose 9 or 8, and could lose 1-6, giving me a win. 3. Ignoring 1-6. I could steal points there. 4. Heavily attacking 4-8 or 5-9: I reinforced 8 and 9 with 26, hoping to beat any 25s there. 5. Relying on multiples of five. I used 6s and 1s to end my numbers, hoping to gain an edge over those who end their numbers with 5s and 0s. "
1,1,1,1,1,1,21,21,26,26,"Re-submitting answer, not sure if it's allowed or not. While debating with friends over optimal strategies, I realized that my initial proposal ([1,1,1,1,1,1,23,23,24,24) was weak to the very ""easy"" solution of 25 in the upper 4 castles. This solution rectifies that mistake."
1,1,1,1,1,1,14,20,20,40,
1,1,1,1,1,1,13,27,27,27,"Trying to guarantee 8,9, and 10. Sending 1 troop everywhere in case other participants don't. Make the troop count sent to 7 be enough to beat someone who sends equal amount to all posts.

Without having any idea of the distribution of possible strategies, this is a complete shot in the dark. Maybe I'm lacking in imagination, but I don't think there is any pure strategy Nash equilibrium to this game. "
1,1,1,1,1,1,10,21,28,35,Fastest to 28 wins
1,1,1,1,1,1,10,19,25,40,7+8+9+10 > 1+2+....+6. Putting 1 at 1-6 will beat all the people who put 0 at those castles.
1,1,1,1,1,1,4,30,30,30,"The winner needs at least 28 points. The quickest route to the win is to obtain the points at the least number of castles possible, which is four: castles 10, 9, 8, and an additional castle (any will do). The obvious strategy is to deploy even numbers of troops at each of those castles. I chose instead to deploy better-than-even numbers at the most valuable castles, but not at the expense of failing to contest every castle, because my goal is to win merely one more beyond the top three. In any situation where I win the top three castles, I win at least one of the others - or tie. 

Fun puzzle; thanks for offering it!"
1,1,1,1,1,1,1,31,31,31,Just in case... Ya know? 
1,1,1,1,1,1,1,31,31,31,"Since you need at least 23 points to win each war, I decided to throw most of my soldiers at castles 8, 9, & 10, for a total of 24 points if my opponent happened to underload those three castles.  I also gave 1 soldier to each of the other seven castles, so if my opponent didn't bother with these castles, I'd win just by showing up."
1,1,1,1,1,1,1,31,31,31,"I just need over half the point to win. So winning 8, 9 and 10 i enough"
1,1,1,1,1,1,1,30,31,32,"Go in at full force to get the 27 points from the most valuable castles, and hope the opponent leaves one of the other 7 undefended"
1,1,1,1,1,1,1,29,30,34,"Need 28 points to win so try to get 10, 9, 8 and then hope to split/win one other castle "
1,1,1,1,1,1,1,28,31,34,"This is a second submission, so please disqualify this one if only one submission is allowed.   I wanted to see how ""dominate the blue states"" faired, so I proportionately distributed to castles 8,9,10  and then removed 7 (2,2,3) to distribute 1 per castles 1-7."
1,1,1,1,1,1,1,21,31,41,A bunch of 1s on lower castles to win points from the presumably many people who submit zeros. Then heavy deployment to the top three castles.
1,1,1,1,1,1,1,1,1,1,"Opponent can't focus on all of the castles, so I'm spreading out to try to win more volume. The low ones aren't valuable but I'm more likely to win those outright."
1,1,1,1,1,0,11,17,26,41,Heavy on big castles
1,1,1,1,0,15,21,0,28,32,"Each point is worth about 2 soldiers. But like gerrymandering, you want to win a lot of castles by a slim margin and lose a few castles by a large margin. So I didn't compete for castles 8 and 5 and hope to use those soldiers to win the other big castles by a little."
1,1,1,1,0,0,0,48,0,48,"28pts wins. I hope my opponent won't play for  castles 1, 2, 3, and 4, and so I put one soldier each there, splitting the remainder between castles 8 and 10 to make exactly 28. Cool idea, BTW!"
1,1,1,1,0,0,0,32,32,32,"Tell us denote a particular deployment by a 10-tuple, castle 1 first.  So the above deployment is (1, 1, 1, 1, 0, 0, 0, 32, 32, 32).

I have been considering 3 broad classes of strategy.  (Obviously there are deployments which don't fit into this schema, but which may still be meritorious.) I call these classes Paper, Scissors and Stone.

Paper strategies cover all the castles with forces approximately proportional to the value of the castle, for example (1, 3, 5, 7, 9, 11,  13, 15, 17, 19).  I also consider an equal distribution of forces, 10 to each castle, to be a Paper deployment.

Scissors surgically target a winning subset of castles, for example (10, 0, 0, 0, 0, 0, 0, 30, 30,. 30).  Clearly Scissors will defeat paper.

Stone strategies target subset of castles insufficient to win on their own, but additionally hope to win or tie enough other castles to gain the extra points to win the war. (1, 1, 1, 1, 0, 0, 0, 32, 32, 32) is a stone strategy.  It will win if it wins castles 8, 9, and 10 and either wins castle 1 or ties any other castle.

Stone loses to paper (it wins its targeted castles but loses the rest).  It mostly wins against scissors because both strategies are likely to contest at least one high-value castle, and stone's forces will be more concentrated.

It's my expectation that the majority of depoyments submitted will be Scissors or paper-scissors hybrids.  My original idea was the stone (1, 0, 0, 0, 0, 0, 0, 33, 33, 33)  This is elegant in that it wins precisely when it wins castles 8, 9, and 10, and any other castle is uncontested by the opponent.  The first condition is nearly certain against a scissor strategy since these must target at least four castles, and it will be very difficult to commit as many as 33 soldiers to any one of them.  The second condition is much less certain.  I cannot predict how many competitors will decide to contest every castle.

I decided to tweak my original idea as I suspect that rather more scissor players will put at least one soldier into every castle than will put exactly 32 into one of 8, 9,  and 10.  I consider it unlikely that a scissor strategy would put more than 1 soldier into castles 2, 3, and 4, without also commuting similar or greater forces to castles 5, 6, and 7.  If he does that, it's not really a pure scissor, more a paper-scissors hybrid."
1,0,5,5,0,24,30,35,0,0,Just get to 28. Leave 9 and 10 alone because they are too enticing.
1,0,4,10,20,26,27,10,1,1,"Assuming most people would enter their highest numbers in castles 9 and 10, I tried to win the lower groups to win the majority of the lower numbered castles"
1,0,0,39,0,60,0,0,0,0,I assume a lot of people are going to go all out on Castle 10. I just am trying to avoid confrontation and maximize my chances of beating people who went all on ten.
1,0,0,1,18,20,20,20,20,0,"Forfeit the richest castle to increase odds at the next five. Figure I'll get some of the 2 & 3 values by ties, but hope for an easy point for the 1 pointer. By my estimation of likelihood of winning/tying each castle, I estimate I'll average 29.25 points out of 55. With a bunch of smart people competing, I figure that's not half bad. "
1,0,0,0,1,15,15,15,20,33,"There were far more points to be gained in the high value castles than in the low value castles, so I weighted my distribution heavily toward the upper end, calculating that if I gain control of the high value casualties, I could afford the loss of the low value castles. "
1,0,0,0,0,0,0,33,33,33,
0,26,32,42,0,0,0,0,0,0,"I thought people would fight it out for the high value targets and end up splitting a fair number of those castles.  If I could take enough of the lower (less competitive castles) I would win more points.  Also, stay away from clean looking numbers."
0,25,25,25,25,0,0,0,0,0,HUNGER!
0,21,0,0,0,0,25,0,27,27,"I think I need to get my minimum 28 points by trying to take 4 castles, which is the minimum it would take. There are 27 combinations of 4 that generate at least 28 pts. All of them require some combo of castle 10, 9 or 8. I chose the one that has  two castles well below 8 where I think there will be less competition and concentrated my troops on 9 and 10. Bit of a punt. Can't wait to see the results!"
0,20,0,20,20,0,0,20,20,0,Guesswork
0,20,0,0,20,20,20,20,0,0,"I need 28 points to win, and any points past 28 are just accessory.  I will send 20 troops each to the castles that will get me to that mark of 28, namely, 8 + 7 + 6 + 5 + 2 = 28."
0,17,12,12,12,12,0,35,0,0,Try to capture 28 exact points (majority) without going for castles 10 and 9
0,16,16,17,0,17,0,17,0,17,"A wild shot in a dimly lit room. Loosing any one of these forts would guarantee victory. By focusing on fewer castles, but allowing for a plan B, I feel like there is a fighting chance. To be clear, this is a guess. "
0,15,14,14,0,14,15,14,14,0,Sacrifice a few castles completely to have a better chance winning against all the even splits. 3 was a hunch but considered only sacrificing 1. Then on a psychology hunch I assumed the ends and the middle would be where people did something screwy so those were the sacrificial castles.
0,15,0,0,20,20,20,25,0,0,It's a race to 28 points. I stacked my castles to focus on a path to 28. 
0,15,0,0,19,20,21,25,0,0,"I assume my enemy will concentrate their forces to take the most valuable positions in order to assure victory, so instead I will concede the most valuable castles to the enemy, and hope to catch them unawares by allocating all my forces to defend the lesser towers to achieve the bare minimum necessary for victory. "
0,15,0,0,18,20,22,25,0,0,Need 28 to win. I put troops on 5 castles to win 28 points. I concentrated my troops on these 5 castles to win them all. I forfieted the 10 and 9 ones as I suspect these will be heavily contested. So hopefully my strategy is unique and successfull enough times.
0,14,0,0,16,20,24,26,0,0,Need to win 28 points to win the fight.  I picked the fewest number of castles required to get 28 points while also trying to avoid the most obvious castles.
0,12,0,0,20,21,21,26,0,0,I chose to punt the highest 2 castles and focus only on 5 castles that would give me a majority. I tried to choose odd numbers that would be less likely to tie for these castles with high amounts of troops.
0,12,0,0,18,20,22,28,0,0,NA
0,12,0,0,0,0,28,0,28,32,"I wanted make sure I was always ~2-3 points above a multiple of 5, since I think a lot of people will use either a multiple of 5, or add 1 extra to a multiple of 5. This is a risky strategy since I only bet in 4 rounds, and I need to win every single one of them. However, I think many strategies will be vulnerable to this one."
0,11,1,1,21,21,22,23,0,0,tried to win 28 points in way that would defeat some of the unimaginative distributions
0,11,0,14,0,19,25,31,0,0,"28 points will win any battle, so any troops deployed fighting for more are essentially wasted. I decided against chasing the top castles, as they may typically require more resource, and focused on the required 28 points on a sliding scale, aiming to take no castle within that for granted."
0,11,0,0,21,21,21,26,0,0,"Need 28 to win. Decided to put all my eggs in one basket: must win 2,5,6,7,8 (unless tie with 0 at just the right spots). "
0,11,0,0,16,17,27,29,0,0,
0,11,0,0,16,16,26,31,0,0,"Need only 28 points, avoiding human friendly round numbers and avoiding high interest rounds. Also playing above 10 to outperform the basic 10x10 strategy."
0,11,0,0,12,15,27,35,0,0,"I went for the minimum number needed to win (28). Guessing that most people might want to try to lock down 10,9,8, I bet more than 1/3 of my total men on 8, then enough on 7 to lock out other competitors, and then enough on 6, 5, and 2 to beat out people just playing the average."
0,11,0,0,12,15,27,35,0,0,"I went for the minimum number needed to win (28). Guessing that most people might want to try to lock down 10,9,8, I bet more than 1/3 of my total men on 8, then enough on 7 to lock out other competitors, and then enough on 6, 5, and 2 to beat out people just playing the average."
0,11,0,0,12,15,27,35,0,0,"I went for the minimum number needed to win (28). Guessing that most people might want to try to lock down 10,9,8, I bet more than 1/3 of my total men on 8, then enough on 7 to lock out other competitors, and then enough on 6, 5, and 2 to beat out people just playing the average."
0,11,0,0,12,15,27,35,0,0,"I went for the minimum number needed to win (28). Guessing that most people might want to try to lock down 10,9,8, I bet more than 1/3 of my total men on 8, then enough on 7 to lock out other competitors, and then enough on 6, 5, and 2 to beat out people just playing the average."
0,10,30,60,0,0,0,0,0,0,swag
0,10,10,10,10,10,15,15,20,0,Random
0,10,0,12,0,20,0,28,0,30,Each even castle is worth more than the odd castle before it
0,10,0,0,25,25,28,12,0,0,"Just hoping I get lucky, if I'm being honest, Ollie."
0,10,0,0,20,20,25,25,0,0,28 to win!
0,10,0,0,20,20,20,30,0,0,"The plan here is to give myself a chance everywhere. There are 55 available points, so 28 points are needed to win. So I divided my troops among 8, 7, 6, 5 and 2 (28 total points), with the assumption that 10 and 9 would be highly coveted, so are best avoided."
0,10,0,0,0,0,0,30,30,30,"I'm ""cheating"" in that I am doing the opposite of my first battle plan.  "
0,10,0,0,0,0,0,30,30,30,"My goal was to defeat the strategies I thought would be most commonly used, specifically, 10 at every castle, 25 in castles 10-7, 25 in castles 10-8 and 25 in 1. My strategy does lose to 10-8 34 33 33 however I don't think that strategy will be heavily employed as it loses to 10 at every castle. "
0,9,11,0,0,19,21,0,0,40,"I didn't work the math out precisely, but I first established that I only wanted to deploy troops to win 28/55 victory points to avoid spreading myself thin. Next, I gave some thought as to which Castles I felt would be least likely to have troops deployed. This is primarily guesswork, but I went with 2, 3, 6, 7, and 10. For the distribution of troops, I put more eggs in the higher values castles but didn't calculate too much beyond that."
0,9,0,5,8,5,27,12,31,3,"I let a computer evolve the strategy. I started with 100 random deployments, then used a Monte Carlo algorithm to develop a deployment that would defeat as many of these as possible. I repeated this procedure until I had a new collection of 100 deployments, each one able to defeat (nearly) every deployment of the original 100. I then repeated the entire process 100 times (100 is a nice round number), each time creating a collection of 100 strategies that were all good at defeating the previous collection. I then selected from these 100 strategies the one that would win when these 100 went up against one another."
0,9,0,0,0,26,30,35,0,0,"We have to keep in mind, our goal is to beat other people, not randomness. My feeling is that most of the analytical riddler minds will modify proportional distribution, giving slight edges to certain castles to try to win them by slight margins, as this seems like the optimal plan. So let's turn that on it's head, and beat a lot of people who smoothly allocate their points. 

There are 55 total points, so 23 total points win. There are many ways to get this with only three castles, but let's keep in mind people will tend to try to do sneaky things to steal high number castles (particularly #9, as that seems ""sneaky"" to ignore 10 and steal 9). 

My first reaction was: just win 7,8,9. Put all your points in and win those. This gives 24 and a sure win. But again, 9 seems like a very highly contested castle. So I decided instead, 6,7,8,2. Surely 2 and 6 should be more guaranteed than 9!

Now just how to distribute. Well, I should mirror how others will be distributing their points here. (obviously 25 troops to each could lose me 7,8 somewhat frequently). While it seems like I MUST win all four to win, many people will likely assign 0 to some castles, so tie points may come into effect. So even losing 6 can be repaired by a tie in 10 and 3.  So I aim to get 23 total points, so let's assign proportionally:

{0,2/23*100, 0,0,0,6/23*100, 7/23*100, 8/23*100, 0, 0} = {0,9,0,0,0,26,30,35,0,0}. 

I need to win every one of these four I've chosen (unless other people elect 0 on some castles... very possible?), but I think in the long run, I've overvalued weird castles that aren't likely to be beaten in general. "
0,8,0,14,0,21,25,0,32,0,"Instead of comparing all options, I compared all combinations that sought to defend 4 castles and promoted the best 10 combinations to an 'a-league'. These combinations were subject to constraints: the total points being defended by at least one army was 28 or more and the armies were allotted to the castles proportional to the number of available victory points for the number of castles I decided to defend. I then did the same for all combinations that sought to defend 5 castles, 6 castles, and 7 castles, 8 castles, and 9 castles. I then ran these a-league combinations (60) against each other and found that this combination won 43 fights, tied 16 fights, and lost none. http://imgur.com/a/TUJmZ. Interestingly, this wins at most 28 points and is thus vulnerable to  0     7     0    14     0    21    25     0    32     1  and the like. I suck at game theory and I'm counting on not everyone coming up with this optimization and having one person specifically beat it."
0,8,0,0,20,22,24,26,0,0,All in on middle sized castles avoiding the high value prizes. Added Castle 2 to get over the half way point.
0,8,0,0,18,21,25,28,0,0,Put all my resources into getting 28 castle points. Hope is to just barely win.
0,8,0,0,18,20,22,32,0,0,NA
0,8,0,0,16,22,25,29,0,0,"Castles 9 and 10 are bound to be hotbeds of WWI-style massive trench wars of wasted troops, numbering in the dozens per side.  I decided to commit all of my forces to only the minimum number of castles that weren't 9 or 10, committing a troop count roughly proportional to the portion of 28 points that that castle provides, hoping to simply win those five and only those five every time, winning by a score of 28-27 very often.  This strategy could massively backfire, of course, if I end up winning four out of those five castles every time, winning no battles overall, but when you're trying to be the top of the heap in a massive free-for-all...go big or go home, and be proud if you land flat on your face trying something crazy!"
0,8,0,0,15,20,26,31,0,0,Figured if I punt on 10 and 9 I should win 5-8. Only need to get to 28 to win. No point going for more than that. 
0,7,10,14,18,21,0,30,0,0,"I submitted an answer before, but mis-added and realized my answer could not possibly win. The strategy here is to maximize my chances at getting 28, instead of maximizing the average outcome. I figure by avoiding the high castles, which most people will probably at least try to bid on, I can ensure I get the 5 lower castles that add up to 28, which is more than half of the total points. The strategies this works well against are the linear strategies (where the soldiers are distributed according to castle size), as well as a top heavy strategy, as long as I can win #8. Most people who play a top heavy strategy will accordingly weigh #10 more than #9 more than #8, so my 30 soldiers on 8 might do the trick. "
0,7,8,10,12,13,15,17,18,0,"I sacrificed 1 as not work it, and 10 as too expensive to fight for. Among the rest, I tried to balance value against risk, to capture the majority of points in the middle."
0,7,0,14,0,21,25,0,33,0,"Ranked by troop efficiency points by expected value, all-in"
0,7,0,0,21,23,24,25,0,0,It takes 28 points to win.  I sacrificed five castles and over-defended five others worth exactly 28.  I figured most opponents will over-defend Castles 9 and 10.
0,7,0,0,18,22,25,28,0,0,If I win email me and I'll explain.
0,7,0,0,18,21,25,29,0,0,"9 and 10 are likely to be prime targets, instead of an arms race there I'll take the likely advantage across other castles needed to win.  This strategy sucks because I have only one path to victory, though."
0,7,0,0,18,21,25,29,0,0,"These five castles are required to finish with one more point than the opponent. This allocation maximizes the value to point ratio (dollars per WAR in baseball?). This assumes opponents will allocate value to the castles that are conceded. If not, the point values will be split potentially adding to the margin of victory."
0,7,0,0,16,16,26,31,2,2,
0,6,7,11,14,17,20,25,0,0,"There are a total of 55 points and I will need 28 to win. I don't have a specific stategy, but I am giving up 20 points by putting zeros on 1,9,10. I am guessing that most people will bet heavily on 9 and 10. I put more weights on 8 as I cannot win without castle 8. Besides that, I will win if I get 6 out of the 7 intended castle."
0,6,0,12,0,23,26,0,33,0,"the maximum point in this game is 55, so to win in any 1 on 1 match up i only need to get 28 points. so focusing on specific castle(s) with total points of 28, i could distribute 100 troops in only 5 castles to guaranteed a control. this strategy can also works if i decide to put 100 troops in 4 castles e.g 1, 8, 9, 10. However majority of people tends to contest ""high-value"" castle(s) (castle 7, 8, 9, 10) so it would be a safer pick if i just distribute the troops in less-contested castles."
0,6,0,0,17,22,26,28,0,1,"Ah, another game of trying to think one level ahead of your opponents. The Meta game is strong this week. The goal is to win small and lose big in terms of the number of troops used. So I plan on losing big on 10, but also beating those that did not deploy any troops. Also, with 55 points up for grabs, getting 28 points is the key to victory. So I will be going after 5-8 and 2, as I expect everyone to try and get either 10 or 9. Then I will be sending 0 troops to the other castles. I did think about trying to send negative soldiers but if I won, I would surely be caught. Good luck to me."
0,6,0,0,15,21,27,31,0,0,If I win the five castles I put troops at I win by one.
0,5,11,17,5,11,11,34,3,3,"Assume most common strategies are to win 1,8,9,10 or 1-7 and counter.  Adjust to defeat 10 per castle also."
0,5,10,10,0,0,20,25,30,0,Need 28 of 55 to win a matchup. 
0,5,7,13,19,26,0,30,0,0,I'm just trying to get to 27 of the 55 points and I thought that a disproportionate number of contestants would fight it out for castles 9 and 10.  But some would think like me and perhaps try only castles 1-7 or 2-7.  So I added 8 and dropped 7 just in case.  
0,5,7,9,11,14,16,18,20,0,Going to punt 10 and 1 and distribute my forces according to value of each castle.
0,5,5,5,5,5,20,55,0,0,"You need 28 victory points to win. I figure most people will will try to win 10, 9, 8 and 1 to get the 28. I'm willing to concede 1, 10, and 9 and want to win the rest."
0,5,5,5,5,5,20,35,20,0,"I figured most people would go for castle 10 to achieve the most possible points, but I believe that you would gain more by grabbing castles 7, 8, and 9. I gave the other castles a few more troops just in case I could grab a few points if those castles were left ignored."
0,5,1,16,22,8,11,11,14,12,"I generated several different strategies, and also created a large number of ""random"" distributions of troops. This was the random distribution that had the greatest success rate against both the deployments I generated myself, and the other random distributions. I slightly edited the random distribution to improve win percentage. "
0,5,0,15,20,20,20,20,0,0,
0,5,0,1,1,1,1,23,28,40,"Winning castles 8,9 and 10 insures 49% of available points. Winning any other castle (other than 1) insures a victory"
0,5,0,0,16,21,27,31,0,0,"As 55 total points are available, 28 are needed for a victory.  Castles 8, 7, 6, 5, and 2 combine for 28 points and will avoid the significant troop commitments likely required to capture castles 9 and 10.  

Gambling that Castle 2 will not be heavily contested does allow for additional troop allocation among castles 5-8."
0,5,0,0,16,21,26,32,0,0,"I need 28 points.  I get split points if my opponents also give 0 to  1, 3, 4, 9, or  20. I chose middle of the road castles to add up to 28, assigning more soldiers as value increased.  Nothing overly mathematical about it. "
0,5,0,0,16,21,25,29,2,2,"100 soldiers/28 points to win = 3.6 soldiers per point if none are wasted. so I focused on 8,7,6,5, and 2 castles for close to that 3.6x ratio. The goal is to win 28 and only 28 points by keying on the marginal value of an additional soldier. The one luxury is that I've allotted for 2 soldiers in castles 9 and 10 to catch others trying to slack off with 0 or 1 there."
0,5,0,0,14,17,24,36,0,4,"Even though this is a zero sum game and can be solved in theory by a linear program, there are (109 choose 9) possible strategies for each player, more than 4 trillion, making the problem computationally infeasible. In addition, there is no pure strategy equilibrium. So we have to develop a heuristic based on our intuition about how we should play. I used the following rules to select a subset of strategies and then picked one at random: 1) The strategies which go after only 4 castles require two of castles 8,9 or 10. Therefore we expect that when these castles are attacked, they will be attacked with large numbers. We only go after one of these three with large numbers. We will use at least 34 soldiers for this castle. 2) A compact strategy (attack fewer castles) avoids spreading the troops to thin. We focus on strategies which only attack 5 castles. 3) A small number of soldiers should be reserved for castle 10 in the event an opponent uses a similar strategy of avoiding the high value castles. 4) The number of soldiers used for the smaller castles we go after should be roughly proportional to their value.

I narrowed it down to 11 castle combinations and soldier assignments and chose the above one at random from them."
0,5,0,0,11,18,26,40,0,0,Didn't go with 10 or 9 because the initial reaction would be to go with the highest points so i started with 8 and went down to 4 given me enough points to win when I add in castle 2.
0,4,6,10,15,25,0,40,0,0,Choose the most unlikely combination to be opposed to score 28 points.
0,4,6,8,10,20,24,28,0,0,Heuristics!!!
0,4,0,2,2,2,23,28,38,1,"The goal is 26 points. The easiest way is with pouring resources into three towers (10,9,7). But then there's no contingency for losing a tower. I'm planning on losing a tower but having enough safety towers to make up the difference."
0,4,0,0,23,23,24,24,1,1,"Trying for exactly 28 out of 55 possible points by winning fewest contested battles (but allowing a small chance of winning 9 and/or 10, when battling people almost like me)."
0,4,0,0,15,21,26,34,0,0,"I first noted that I only needed 28 points to win the war. I then started finding all of the combinations of castles that add up to 28. I liked 2, 5, 6, 7, 8 the most because I felt that I was making my opponents waste troops on castles 9 and 10. I thought this would leave the middle castles open for my taking, while hopefully leaving just enough behind to secure castle 2."
0,3,6,13,0,4,20,27,26,1,The best result from a quickly-bodged genetic algorithm. If I realized the deadline was EST this is what I would have submitted then.
0,3,6,11,14,17,18,19,11,1,
0,3,5,1,2,14,12,28,1,34,"I ran a series of simulations, using a sort of fuzzy genetic algorithm. I started with thousands of random sets, saw what succeeded, and began to apply mutations to the successful sets. Through many hundreds of mutations, certain strategies started to emerge. None appeared drastically stronger than the others. So I chose one of the stronger mutations and am curious to see how it does. : )"
0,3,4,7,16,24,4,34,4,4,"The idea is to win forts by small amounts and to lose by big amounts.  Am thinking most people will heavily attack castles 9 and 10 which I hope to lose by big amounts, while I hope to win castles 2, 3, 4, 5, 6, and 8 for a 28 to 27 victory!"
0,3,4,3,10,15,15,22,20,8,Many simulated troops died to bring us this information.
0,3,4,1,0,1,22,24,21,24,"Need to win at least 28 points to win a battle. A reasonable strategy should be to exceed the expected number of soldiers at each castle if they were evenly distributed according to points such that you gain at least 28 points (e.g. try to win castles 10, 9, 6, 3). In practice, I used a genetic ""like"" algorithm to randomly evolve a population of 1000 strategies that competed against each other and took the best performing strategy out of that population after 1000 generations. The algorithm used elitism where the top 10% of strategies were carried over from one generation to the next. The top 40% of strategies were each randomly modified by shifting a few soldiers around (15 on average). The final 50% of the population at each generation was a set of random new strategies."
0,3,3,7,6,8,11,15,20,27,"Weighted according to the square of the value, because wasting resources at a castle I don't win is the worst outcome.
Introduced some random variance (10% of the armies) so this couldn't be easily gamed with a ""one more at each, way less at one"" strategy. This was also distributed according to the squares of the value.
0    3    3    7    6    8   11"
0,3,3,3,21,0,21,24,1,24,I wanted to win 5 regions
0,3,3,3,17,3,31,35,3,2,"Need to get to 28 points.  Focus on 3 to get me to 20, then deploy a few everywhere else to pick up 8 points on poorly defended castles."
0,3,3,3,3,20,20,20,25,3,
0,3,0,0,16,24,26,31,0,0,"Not sure if we are allowed to choose 0 for any but I figured all you need is to take castle 5,6,7,8 and then 2 and you're there. "
0,3,0,0,14,17,19,22,25,0,"I am conceding castle 10 completely, since I expect others will attack it heavily. I am also conceding castles 1, 3, and 4 based on their low point value.  I did defend castle 2, since sending troops there means I could lose one of the six castles I sent soldiers to and still have the most points in the battle.  Then, I distributed my 100 troops among castles 2, 5, 6, 7, 8, and 9 weighted on the points available, rounding up all fractions for castles 5, 6, 7, 8, and 9, which leaves only 3 left for castle 2."
0,2,4,6,8,14,15,16,17,18,"I wrote C++ code to randomly assign soldiers several million times, then tweaked the best result a bit to get marginal improvement and make the values flow better."
0,2,4,6,8,12,14,16,18,20,"Win the bigger castle over uniform strategists, while not getting blitzed too much in the smaller castles"
0,2,3,8,10,12,15,20,30,0,Random
0,2,3,3,6,17,17,17,17,18,Private algorithm...
0,2,2,11,20,2,2,2,28,31,
0,2,2,5,6,10,12,17,20,26,The troop numbers are proportional to the square of each castle's value. Let's see if this works!
0,2,2,2,2,14,21,26,31,0,"My first thought was to concentrate all of my forces on the minimum number of castles, so as to maximize my chances of winning each one. The minimum required to win the war are castles 7,8,9 and 10, which give me 34 out of the total 55 points in the game. Then I realized that everyone else would likely be doing something similar, while also probably assigning more soldiers to higher-valued castles. This however leaves all the lower-valued castles up for grabs, so it's better to completely forget about castle #10 (which would probably need the most soldiers in order to be captured) and assign just a few soldiers to castles 2-6. I went for a minimum of 2 soldiers per castle, thinking that my opponents would probably use a minimum of 1 (or 0) for castles they don't value much."
0,2,0,9,4,1,20,0,28,36,"I generated a few hundred random teams, tested them against a few base deployments (10 in each castle, 11 in castles 2 - 10, etc) and each other, and this one appears to be the best."
0,2,0,2,22,0,26,0,12,36,Simple simulation among randomly generated strategies https://github.com/mattinbits/fivethirthyeight_riddler_war_game
0,1,13,14,14,14,15,14,14,1,"Happy to lose 1, 2 and 10. Leaves 43 winnable points. Need 28. So expect to lose 9 or 8. Expect to win 3-6 which gives a solid shout! Better than an all 10 strategy! Just sneaks the almost double the value strategy"
0,1,11,2,11,3,3,23,5,41,"Determined combinations of castles that would give you 28 points, optimized for ""cheapest"" castle. Then developed an algorithm that distributed points to those castles in a variety of ways, as well as general approaches to the problem -- distributing points evenly, distributing points across all castles as ratios of value, etc. For each run of the script found the winner, then manually added approaches that looked like they could beat the winning combination, and did that until I couldn't improve it anymore."
0,1,10,13,2,20,23,27,2,2,"28 points wins a round.  This attempts to find a reasonably efficient path to those 28 points 8, 7, 6, 4, 3.  This avoids fights with any top heavy strategy except perhaps on the 8.  A few troops on other castles is intended to capture full points on any castle zeroed out by an opponent."
0,1,7,10,10,18,23,28,1,2,"Castle 10 and 9 are going to be bloodbath. Just putting 3 soldier to gain all the people like me who are not going to try fighting them. I can lose either 3-4-5 but not both, so similar amount in all of them. "
0,1,7,4,8,5,1,21,23,30,I made a bunch of random strategies fight and chose the one that won. I really didn't have any idea how to pick a good strategy.
0,1,6,8,15,2,19,21,26,2,"Step 1. Allocate troops proportional to value +6. (160 total)
Step 2. Surrender castle 10 and 6 to avoid bloodshed in favor of fortifying other castles. Leave 2 troops in 10 and 6 just in case. (123 total)
Step 3. Reduce forces in castle 1-4 to comply with 100 force total.
Step 4. Randomly reallocate to counter human tendencies."
0,1,6,8,10,11,13,15,17,19,"(100. / sum(np.arange(11)))*np.array(np.arange(11))
Then rounded up."
0,1,6,8,10,11,13,15,17,19,"There are 55 total victory points.  On average, each point will be secured by 100/55 of a soldier.  Trying to allocate that number of soldiers to each point gives an approximate number of soldiers to allocate for each castle.  I took the ceiling of each number to ensure that I sent at least enough soldiers to capture the castles based on their worth.  Rounding allowed exactly 100 soldiers to be allocated to each castle, but on average, I would be losing the 10 and 9 castle to be able to win the 1 and 2 castle.  Not a good trade-off.  As a result, I had 5 extra soldiers allocated.  I simply took them away from the lowest valued castles (2 from castle #1 and 3 from castle #2) until I had gone back down to 100 soldiers.  The most tempting soldier to redeploy is at castle #5 since I think I need 9.09 soldiers and have allocated a full 10.  That is a lot of wasted soldier, but is worth securing castle #5 over getting a small chance at castle #1 or #2."
0,1,5,5,5,0,21,21,21,21,
0,1,4,5,6,9,14,14,22,25,Ran lots of Python simulations. Winning strategies tended to have about a quarter of the troops at Castle 10.
0,1,3,6,9,13,17,22,28,1,"Firstly, I decide not to seriously contest castle number 10. As it's a high value target it can be expected that many people will throw a lot of soldiers at it, and without contesting it I'll have more soldiers left over to have a better shot at claiming the other castles.

Now I want to assign troops to the remaining castles, but ensure that I assign more troops to the more valuable ones. I square the value of each remaining castle (so castle 3 is worth 9, castle 9 is worth 81, for a total of 285 points) I then assign troops to each castle with a weighting proportional to this squared value (so castle 9 gets 81/285*100 soldiers, or 28) Due to rounding errors there's one soldier left over after this process, I place him in the 10th castle just in case my opponent has also decided not to contest it."
0,1,3,6,1,23,2,27,35,2,"Gameplan: Win 28 victory points through capturing castles. 
Objective: Conquer the kingdom.

The total value of the castles is 55. The total victory points needed to win is 28, assuming no ties. The first thing to do is to establish the minimum number of ways to reach the winning number, given some assumptions and assessing potential strategies.  

I had to be careful that my strategy could not lose to an even distribution, and I assumed that everyone else would take the same precautions. Now, I wouldn't have to worry about other strategys that would lose to 10 troops at each castle, 20 trooops at 5 castles, etc. What I did need to worry about were more developed strategies like a bell distribution and whether or not people would put more than a third of their troops on one castle to secure a victory. 

Initially, I believed that giving up free castles was a poor strategy, so I started with the idea that a troop must be sent to each castle. That most players would send most of their troops to castle #10 was an assumption I kept. Assessing the average of winning #8 and #9 was as good as winning #7 and #10, I decided to lose the a few battles, and win the war. 

After many campaigns, I had a lot of rock/paper/scissors where one of my strategies would lose to one that didn't follow assumptions, but that one would lose to a more deliberate bell-curve or even spread. I decided to remove the requirement of challenging each castle, and kept optimizing, while ensuring they were safe from easily preventable defeats. "
0,1,3,4,5,6,34,36,5,6,"I am hoping that by fortifying 7 and 8, I can ruin the strategies of people hoping to win a minimum number of high value castles (4 is the minimum number of castles you must win). By throwing slightly more than a minimum number of soldiers at 9 and 10 I hope to have dedicated enough so that people that wanted them wasted way more soldiers then they needed and people that just committed as an after thought would lose because they didn't commit enough."
0,1,3,3,11,18,25,33,3,3,"Went in heavily for 8,7,6, and 5 while keeping some forces at 10,9,4, and 3 in case opponents put a weak force there. This alignment won head-to-head the most often against a lot of other strategies that I considered. Any possibility that you are able to publish the complete head-to-head records of all participants? Thanks, this was a fun one (and the first time I've formally entered!)."
0,1,3,3,6,10,20,19,1,37,Lots of computer simulations... then an itsy-bitsy tweak to guarantee I'd beat my own answer.
0,1,3,1,11,12,12,19,22,19,I don't have a good explanation for this. 
0,1,2,11,11,13,26,2,31,2,"My first thought was to come up with a combination of numbers that meets or just exceeds 28, focus on those, and ignore the rest. I suspect many other people are thinking this too. I realized that if I'm too focused on getting the minimum necessary to win, I can be undone if someone spikes one castle I'd counted on. I also didn't want to have a ""backup plan"" though because that only makes my main plan weaker. So my main plan now is to defeat my first main plan, I think I'll win against virtually everyone whose strategy is to win exactly 4 or 5 castles including 7 or 9 (which I figured would be easier to spike than 8 or 10).

I also sent small brigades to some castles I'm not counting on because the risk-to-reward ratio is so good."
0,1,2,7,11,15,19,21,22,2,"Value difference between 9 and 10 not that much, and assumed most people would focus on 10, so shifted resources away. Still wanted a few soldiers in most castles in case people consolidated too much. Ran some crude tests to see how this distribution compared to others like a even, proportional, and various versions of my strategy."
0,1,2,4,7,9,13,17,21,26,"i wanted to slightly beat what i thought the popular solutions would be:
2, 4, 5, 7, 9, 11, 13, 15, 16, 18 (a proportional solution)
10, 10, 10, 10, 10, 10, 10, 10, 10, 10 (a naive solution)
and 20, 20, 20, 20, 20, 0, 0, 0, 0, 0 (concentrating on bigger towers)
i decided to make my armies proportional to the squares of victory points.
these are roughly in agreement with the power index with these voters."
0,1,2,4,7,9,13,17,21,26,Proportional to x^2
0,1,2,4,7,9,13,16,21,27,Weighted the soldiers by the sum of the squares to increasing weight of the more valuable castles 
0,1,2,4,6,9,13,17,21,27,"Proportional to square of castle number.  This beat a number of other strategies I thought of including constant, exponential, linear and a few other powers."
0,1,2,3,4,5,10,15,25,35,"Need 28 or more points to win scenario, most points concentrated in castles 10-7 in that case. Capturing 3 of these 4 gives the most probability of getting 28 or more points."
0,1,2,3,3,16,11,14,42,8,"I started with an even distribution of 10 armies by 10. Then I thought of an distribution that would defeat that setup.  Then I thought of a distribution that would defeat the first two. And so on until I had a distribution that would beat about a dozen other distributions I had thought of and that covered a few different strategies. 

Then I decided to let a computer do the heavy lifting and wrote a script that would randomly modify a distribution one soldier at a time. If the resulting distribution was able to beat or tie all the existing arrangements then it was added to the list to be tested against. 

This distribution is the champion of that process. "
0,1,2,2,4,4,10,25,26,26,generating options for different strategies and selecting the best between them for a small generation.  (pseudo genetic algorithm GA just one generation)
0,1,2,2,2,11,22,25,33,2,"Give up 10, focus on 6-9"
0,1,1,17,1,1,25,26,26,2,
0,1,1,15,1,2,20,23,35,2,"I built a winning route to 28/55 points, but with some degree of irrational placements to beat other people using the same route. Then, I assume a ton of people would put 0 points on some spots, or 1 as a token, so I used a few troops to shore up those in the even someone aggressively using my route failed to do so. "
0,1,1,11,3,3,22,26,30,3,"My gameplan was getting to 28 points with the fewest castles possible. That means 4 castles, with a bunch of different ways to accomplish 28 points. I'm taking the gamble that most opponents will commit significant troops to Castle 10, so I'm more or less ""punting"" that one. That drastically reduces my paths to 28 points to just three: Castles 7, 8, and 9 with one of 4, 5, or 6 added on. I decide to focus on winning 4, 7, 8, and 9, with the idea that Castle 4 *should* be winnable with fewer troops than 5 or 6. Obviously, this allows for nominally more emphasis to 8 and 9. To defend against an opponent using a similar strategy, there are still minimal deployments to Castles 2, 3, 5, 6, and 10 in order to split or steal those Castles from anyone also punting them (and working from the assumption that *most* opponents will punt with either 0 or 1 troop, and maybe a few with 2), which ought to give me multiple outs to catch 28, even if only 3 of my 4 anchor Castles are won. We'll see..."
0,1,1,11,1,1,24,26,35,0,The basic idea is to concentrate forces on a limited number of castles which together award enough points to win the war. 10 is dropped because too competitive. 9+8+7+4 is just enough to win a majority of points. Some isolated soldiers are sent to additional castles in case it with allow some useful gains over opponents having chosen to ignore entirely these castles.
0,1,1,10,11,16,24,34,1,2,
0,1,1,9,11,13,16,29,19,1,Focused on winning the middle castles
0,1,1,7,1,25,30,35,0,0,"There are 55 points at stake, so we need 28 to win.  While the optimal strategy likely involves some randomness, let??s go with a simple deterministic strategy (looking at the submission form it??s deterministic strategies only).  A lot of people will target the high numbers, so we??ll pass on those -- hopefully most people will allocate a lot to 9 and 10 and we can make that up elsewhere.  Since we have 100 soldiers to use to get 28 points we need a return of slightly over 1/4 points per soldier.  Some people might use this as a benchmark, so for the larger numbers let??s overshoot this (because losing one of those battles dooms us).  Rather than pass completely on some of the battles we don??t intend to win (namely 5) let??s throw out a single soldier for a chance at great point per soldier value.

Pts    Sldrs    Ratio
8    35    4.375
7    30    4.286
6    25    4.167
5    1    0.2
4    7    1.75
3    1    0.333
2    1    0.5

We need to win those top three battles to put us at 21, leaving us with 7 more to get.  Winning 4 puts us at 25, and we then need 3 total points between all of our 1 and 0 soldier battles.

We want to be able to match up against some simple strategies, so let??s consider a few.

10, 9, 8, 1 with most soldiers at first three -- Seems like we should beat it since we have more than 1/3 of our soldiers at 8

10 soldiers to each castle -- We lose with 21 points, but I don??t expect this to be common

Approximately 2x at each castle -- We probably lose, depending how the low ones fall

Ignoring 8, 9, 10 -- Depends on allocation, but if we take 6 and 7 we win after splitting 9 and 10.  If we lose 6 or 7 then it will be close, depending on 4

All even (or odd) numbers -- Depending on allocation this could go either way"
0,1,1,2,10,11,31,41,1,2,Other players are likely to target the highest point value castles most heavily.  By targeting mid tier castles it is possible to get to 28 without having to compete for the highest value castles.  1-2 troops are sent to the remaining castles to pick up any that are uncontested.
0,1,1,2,1,13,1,33,31,17,"I coded an evolutionary sim to test different allocations, seeding it and making it compete against both random strategies and some likely 1st- and 2nd-order strategies.  The above allocation was one of a few that seemed to do well."
0,1,1,1,16,16,16,17,16,16,"The idea was to stock up as many soldiers in six castles as possible and have at least one soldier in the other four - so as to gain any easy available wins (in case the opponent goes with a strategy of assigning no soldiers to certain castles. The initial castles are sparsely populated in anticipation of the archenemy stocking up troops en masse in those, and leaving later castles thin."
0,1,1,1,15,18,30,34,0,0,"There's a limited number of possible winning combinations, and with scarce troops any deployment needs to be concentrated in one or two places. Based on possible allocations, this deployment interferes with as many other deployment styles as possible."
0,1,1,1,2,20,20,20,30,5,"Winning on 6 through 9 is enough to win. I expect most people to go to 10 in force, but I'm sending a small force to not disrupt my other plans but still win if the opponent leaves it almost completely empty.
On 1 through 5, I'd rather send a 1 force than nothing, I expect those 5 soldiers to turn up a couple of points several times. At the last moment I chose to send 2 to 5 and none to 1, hope it works."
0,1,1,1,2,15,26,26,26,2,This strategy focuses on castles 6-9. 
0,1,1,1,1,8,13,20,25,30,Back of the envelop guessing plus a little gut feeling
0,1,1,1,1,1,25,23,23,24,"The only way for someone to beat me is if they only deployed their troops to the 4 most valuable castles. and then only if they spread them out evenly. Any over massing of troops means I win. I also capture the any castle that they don't send a single troop too. 

Follow Me!"
0,1,1,1,1,1,1,15,38,41,
0,1,0,1,2,2,1,1,1,1,idk
0,1,0,1,0,1,0,1,0,96,Maximize my chances of winning castle 10 while hedging in the event I lose castle 10 that I get other castles to sufficiently win the game.
0,1,0,0,19,22,26,30,0,2,"I found the average number of troops at each castle necessary to win ""proportional points"" =(Castle Number/28)*100. At each of my ""key castles"" (5 through 8) I put more troops than expected to win those number of points. I guessed people would overvalue higher numbers (i.e. 9 and 10) so I started my firewall at 8. If I win 8 through 5 I will likely tie at some lower castles and get to a win at 28 points. I threw two troops at 10 to pick up against people who abandoned 10 entirely (my own strategy taken to the extreme.) I also threw one troop at 2 - which can allow a clean win if I pick up 8,7,6,5 and 2."
0,0,30,26,20,0,14,10,0,0,"I tried to target 28 points in places that are not as likely to be contested, allocating more troops to the more contested locations."
0,0,15,18,0,20,22,25,0,0,"If I am able to win all 5 that I try for, I will win."
0,0,15,15,15,15,20,20,0,0,random
0,0,15,15,15,15,0,0,0,40,It beat my previous strategy 
0,0,15,1,0,0,0,27,28,29,"There are 55 points up for grabs. I need at least 28 of those points. 10+9+8+1=28. So I need to win 10,9,8, and some other number. 3 seems like a good other number to win. It's probably not too popular. But one in 4, just in case it gets left open and is the points I need to win, or someone puts only 1 in it and it's the points I need to tie. The 27/28/29 beats this exact strategy except with 28/28/28 instead. And it doesn't leave me too vulnerable to 27/27/27s because if they let their fourth number be something other than 4 and they do put 1 into their other numbers, I win. (I win the 3, then I win with the tie from 4). If they don't put 1 into their other numbers, their fourth number needs to be 3."
0,0,15,0,0,20,30,35,0,0,"There are 55 points to win, I need 28.  I don't need to win every round, just the majority so if I deploy to win exactly 28 points in some of the less likeable castles, I should win more often then not."
0,0,15,0,0,15,0,0,35,35,It beat my previous strategy 
0,0,14,16,20,20,16,14,0,0,"The higher numbers are likely to draw more troops from one's opponent, thus making them harder to win. The lower numbers are easier to win, but provide less value per troop. It seems making a play for the middle values offers the highest likely return on troop investment. Since 28/55 is needed to win, the 33 points offered by the middle six numbers would be sufficient to win."
0,0,14,15,0,15,15,39,1,1,"The most general strategy for defeating ""random"" deployments is to pick a set of castles representing a majority of points.  Most obvious would be the high-point castles, and in fact if you look at the 27 combinations of four castles that add up to 28 points or more, each of the top three castles are required for at least 17 of the 27.
So, we expect most strategies to rely on two or more of the top three castles plus two others (25,0,25,0,25,0,25,0,0,0). The available approaches to beat these baseline strategies are:
1) Claim two of those three with overwhelming strength and pick two more with sufficient strength to pick them up against token support (0,37,37,0,13,13,0,0,0,)
2) Figure that almost everyone wants to use either the 9-point castle or the 10-point castle and overload that, then spread the rest fairly widely expecting to pick up the holes in the opponent's broken strategy (1,51,8,8,8,8,8,8,0,0)
3) Execute the strategy more or less directly, trying to claim three of the top four with strength, then choosing a fourth castle to claim with less than overwhelming support. (1,26,26,26,1,17,1,1,1,0)
4) Pick a strategy that requires winning five castles that do not include the top two. (1,1,41,14,14,1,14,14,0,0)
Of these, the third seems the weakest--the others break it. Counterintuitively, the fourth strategy is the most successful. Going hard after the 8-point castle leaves enough points to pick four others against token support that other strategies can afford after preparing to win one of the top two castles. The disadvantage is that it fails against even support 
Variations include how many troops to send as tokens to the other castles, hoping to pick up all of the points from an undefended castle."
0,0,13,13,1,20,23,26,2,2,"The numbers I need to hit to win a given battle is 28.   I decided to try to get it by getting 3, 4, 6, 7, and 8.  I distributed most of my soldiers proportionately among these numbers.   I'm hoping a lot of people will put too much stock in the top two numbers and I will have a better chance at winning these.  However, putting all of my troops in these numbers would make the others winnable with a single troop.   

Ultimately, I decided to forfeit 1 and 2 so that I would have a distribution that would beat someone who put 10 in each or 11 in the top 9 or 12 in the top 8."
0,0,12,16,20,24,28,0,0,0,"Since the sum of 1-10 is 55, it's really a race to 28 points. I figured most people will try to evenly distribute their answers, sacrificing the guarantee of winning a 9 or 10, in order to ensure they *could* win something. I figured I should stack all my marbles to try and guarantee winning 4-5 castles equalling 28 points, which would do well against a balanced distribution. I figured 3-7 was the best way to do this, because people will still weight their answers towards the higher numbers, so 3-7 is a nice sneaky way to ensure I get all of them (if I put 28 on the 10 square, there is a good chance someone with a more even distribution could still beat me on that one). "
0,0,12,16,11,0,14,0,47,0,"I wanted a strategy that would defeat the following strategies:

(1) Maximize points -- give castle N 100N/55 soldiers
(2) Greedy ""maximize chance of getting 28 points"" -- putting 100N/28 soldiers in castles 10, 9, 8, and 1
(3) Basic implementation of my strategy -- 100N/28 soldiers in castles 9, 7, 5, 3, and 2

Versus a strategy that puts soldiers in every castle, like (1) above, my strategy can only get 28 points max. That means I need to have at least 1+ceil(100N/55) soldiers in every castle I try to claim.

Against (2), I need to make sure I win at least 1 of the castles they contest. I decided to contest castle 9, so I'll put my spare troops there.

Against (3), I'll need to win castles adding up to at least 15. I'll probably win 9 with all my spare troops there, so I need to pick more castles that add up to 6 or more and give them extra soldiers. I put 1+ceil(100N/28) troops in castles 3 and 4, leaving 1+ceil(100N/55) in castles 5 and 7.

Final results:
Castle 3: 1+ceil(100*3/28) = 12 soldiers
Castle 4: 1+ceil(100*4/28) = 16 soldiers
Castle 5: 1+ceil(100*5/55) = 11 soldiers
Castle 7: 1+ceil(100*5/55) = 14 soldiers
Castle 9: the other 47 soldiers

Let's see how it goes!"
0,0,12,13,0,22,25,28,0,0,Who cares. It's probably wrong :) I know that I have to win every castle that I put troops in. I just thought about the different configurations that people might choose and I'm hoping that I win against more than anyone else 
0,0,12,13,0,20,25,30,0,0,"Picked castles that, if I won, would give me one more point than those points attributed to castles I lost."
0,0,12,0,0,23,1,1,29,34,Please send me results if and when available. Much appreciated. 
0,0,11,15,0,20,26,28,0,0,"Goal is to get 28/55 points and no more.  Anticipating what other peoples' common strategies might be, I selected a combination of castles and troop deployments designed with an attempt to win exactly 28 points in a typical one-on-one match-up."
0,0,11,14,0,21,25,29,0,0,"Find the total points (55) then minimum winning total (28). Determine which combinations of castles yields 28 points (15 total combinations). Find the percentage each castle would make up a victory condition. Add each percentage of a combination and remove any combination above 100% leaving 6 combinations. Find each castle's percentage of total points then remove any combinations above 50% leaving four combinations (10-8-6-4, 10-9-6-3, 8-7-6-4-3, and 10-9-4-3-2). Went with gut in decision of 8-7-6-4-3 combination due to belief that majority of people will go after 10 and 9. Used percentages from winning total calculations as number of units in each castle. "
0,0,11,14,0,20,24,29,1,1,"You need 28 points to win so only play for them, I went for the mid values as I more people would go all out for the high values."
0,0,11,13,0,22,25,29,0,0,Heavily stacked castles summing to 28.
0,0,11,12,14,16,2,2,41,2,"With a sum of 55, a score of 27 wins, and the top 3 castles sum 27, so the easiest path to victory would be to win all of the top 3, and any path to victory includes one of them. I assume most people will try hardest to win the top three, but that no one is likely to go over 40 for castle 9 (though they might for 10), so with a deployment of 41 I can take castle 9 in most matchups. Then I want to win the magic 27 at the lowest-scoring castles possible, to maximize my chance of victory, which means aiming to win 6, 5, 4, 3. This is where I put the majority of my troops, more in the higher-ranking castles. If I don't win these five (9,6,5,4,3), winning one or two will do nothing for me, so they get none. I do want to have some minimal presence in case opponents totally and completely ignore 10, 8, or 7, though in going after any of the 5 I'm targeting, so 2 will beat anyone who leaves just 1 or 0 in those three spots. "
0,0,11,12,0,16,30,31,0,0,"Cede 9 and 10, concentrate power on 6,7,8 with remaining forces allocated to ensure a point victory is possible"
0,0,11,11,11,21,21,21,0,4,My thought was that most people would go for the highest totals so I would take advantage by going for the middle numbers 3-8. I also figured many people would pick multiples of 5 so going one above would be good. Additionally I had 4 troops left and figured enough people might forgot the 10 in thoughts other people would use their troops their so I put my last four on the 10.
0,0,11,11,11,11,11,0,34,11,"Beats some of the simpler strategies, and is beaten only by strategies with more obvious flaws."
0,0,11,11,1,16,26,33,1,1,"Basically picked a strategy that I thought would beat most simple solutions that people would think of immediately, beat some more complex well thought out strategies, and beat basically no strategies of people with programming to run every possible combination that identifies which solution  wins most frequently against other strategies. This particular warlord has no such modern technology, plus my war-manager has instructed me to get back to work..."
0,0,11,11,0,21,26,31,0,0,I need 28 points to win. I m gonna put everything in castle 8 7 6 4 and 3. I need to win all of them though (can't tie).
0,0,11,11,0,21,26,31,0,0,"If I ""forfeit"" some battles, I can focus my forces on the battles I choose to take.  I can feasibly win with four battles if I take castle 9 and forfeit 10, but I could instead to forfeit 9 & 10 and win with five castles total: 8, 7, 6, 4, 3 (one might also replace ""...4, 3"" with ""...5, 2"").  To win with 6 castles, forfeiting castle 6: Castles 8, 7, 5, 4, 3, 1.  Another option is to forfeit castle 7 as well, again winning the war with 6 castles: Castles 8, 6, 5, 4, 3, 2.  Lastly, if I wanted to win with 7, I'd need to win all the castles from 1 to 7.  These are somewhat minimalist answers, as I tried to forfeit the highest-valued castles possible.  I chose to go with castles 8, 7, 6, 4, and 3, but I tried to avoid multiples of 5, since I suspected them as likely answers from other submitters, and ties on castles 6, 7, or 8 should result as a loss for my battle plan.  
0 on 10, 0 on 9, 31 on 8, 26 on 7, 21 on 6, 0 on 5, 11 on 4, 11 on 3, 0 on 2, 0 on 1."
0,0,11,0,14,18,23,34,0,0,"I'm only trying to win 3,5,6,7, & 8"
0,0,11,0,0,0,26,27,0,36,"This approach goes all-in on winning just enough points to win. It is very vulnerable to ""random"" distributions, which only need to win one of the castles I actually allocate troops to, but puts enough troops in the ""target"" castles that I should be able to win them most of the time."
0,0,10,20,30,40,0,0,0,0,"Went for middle of the road, figuring most would deploy larger troops at the higher values"
0,0,10,15,0,20,25,30,0,0,"I need to get to 28 points and distributed them on 3,4,5,7, and 8 giving up the other 5"
0,0,10,13,0,22,25,30,0,0,"I decided to commit all of my troops to castles 3, 4, 6, 7, and 8. If I win these 5, that will be enough points to win. This beat many of the other things I tried."
0,0,10,11,12,18,23,24,1,1,"I think that most people will try to get the 9 and 10 castles.  I'm giving those away, and trying to get all of the middle ones.  I tested 10,000 random strategies, assuming that 90% of the people will weight toward the top.  Sadly I expect that The Riddler people are smart...and that my goal will fail, but it's worth a shot."
0,0,10,0,21,22,23,24,0,0,"Relinquished 10s, in the hope of winning the middle ground."
0,0,10,0,20,0,30,0,40,0,"Higher points are worth more, so have more troops."
0,0,10,0,0,22,0,0,32,36,"This deployment was optimised to contain the fewest castles required to reach the minimum needed points to win (28). Specifically, I wanted the deployment to have the most ""bang for your buck,""  and to that end I looked for the most efficient castle. The metric I used was troops per point per point, which produced castle number 3, leaving only 2 selections for the castle, 3, 6, 9 and 10 or 3, 7,8 and 10. I chose the combination with the fewest points per troop, and then weighted troop placement by the number of troops I'd expect someone who had placed them based on value alone would have placed them. "
0,0,10,0,0,0,0,30,30,30,"Going straight from either end, leads to Castle 7 being the swing castle. I am therefore avoiding Castle 7 altogether. Putting 25 on the last four seems strong, but you would only have to lose one castle to lose. Putting 40 on Castle 7 and 10 on the first six would be good unless someone won castle 8-10 and stole a single other castle as 8-10 gives 27 points and to win you need 28. To avoid these scenarios, you could take 5,6,8,9 with force, leave 7 alone and try to benefit from a possible 0 on 4+10. Putting 30 on 8-10 and 10 on 2 would win all. Except Castle 2 may not be that under the radar so instead I will go after Castle 3 if that means I win against similar strategies to mine that choose Castles 1 or 2. I am avoiding the middle because of those that will go for the averages, avoiding the high value targets and the worthless low ranking castles. Scenarios that lose would be 31-33 on 8-10, but those would also lose to other scenarios."
0,0,10,0,0,0,0,27,30,33,"A good strategy needs to achieve a number of goals:

1) It should deploy troops in proportion to the number of points to be won
2) It should concentrate on getting enough points to win, rather than trying to win all of the points
3) It should be robust against opponents who make small deviations from the same strategy
4) It should beat every obvious strategy

There are 55 points available but only 28 points are required to win. Therefore, most of the troops are concentrated on winning castles 8, 9 and 10, for which 27 points are available.

The remainder of the troops are concentrated on trying to win castle 3, rather than trying to win castle 7. This is because castle 7 will be more competitive than castle 3, and deploying a proportionate number of troops to win castle 7 will be a waste of resources which could be used to fight for the other castles.

This strategy is robust against another strategy which leaves a lot of the smaller castles undefended. Even if it lost castle 9 or castle 10 to such an opponent, it would still win because of the split points at the castles ignored by both sides.

It would lose to a strategy which attempted to win castle 1 rather than castle 3 but it has an advantage over the latter strategy in that it would beat the ""obvious"" strategy of putting 10 troops on each castle, while the latter strategy would not."
0,0,9,9,9,15,25,30,0,0,"Decided that castles 3-8 were a pretty solid sum of points. 1 and 2 were skipped because they weren't that many points...9 and 10 were skipped because they were more likely to be contested. Incremented the value of troops somewhat haphazardly based on how I thought it should be, because damnit I'm the warlord and can do what I want. I also intentionally kept 3 troops at home because I will need protection when all of my troops die in this bloody war and I need to flee the land.

 I tried to look into game theory formulas to figure out the optimal strat but didn't find anything too helpful. Interested in what other people do. "
0,0,9,3,6,5,19,26,29,3,"I tried to ""grow"" a solution using a genetic algorithm. Turns out that's not the best strategy, since the function you're optimizing depends on the population of solutions you're testing. Still, it was a fun thing to try and even kind of stabilized on a set of strategies (everything ignored 1-2 to some extent and either went hard on 10 or just sent a few there)."
0,0,9,1,1,16,1,1,34,37,"I'm shooting to win castles 3, 6, 9, and 10 for a total of 28 points.  I'm also putting one soldier at 4, 5, 7, and 8 in case there are easy points to pick up there in case I loose one of my preferred castles."
0,0,8,12,15,18,21,24,1,1,"By ignoring castles 1 and 2, and only investing 1 troop in castles 9 and 10, i am effectively conceding approximately 2/5 of the points with the strategy of sending overwhelming forces to the remaining castles with 3/5 of the points. I have invested 1 troop in castles 9 and 10 in order to counter a similar strategy - ignoring the highest value castles - and potentially splitting or winning points a large number of point with only 2% of my troops invested. No maths/game theory  involved "
0,0,8,11,19,22,0,0,0,40,"A proportionate distribution across one combination of must-win castles for the minimum number of points to win. Then less a few soldiers from the lower point castles and re-allocated to the higher value castles, which was guesswork."
0,0,8,11,15,0,27,0,39,0,"Rock-paper-scissors logic:

A ""wide"" strategy that contests all 10 castles (55 points, avg 1.81 men-per-point) will always lose to a ""tall"" strategy that contests barely enough castles to win (28 points, avg 3.57 men-per-point). With a nearly 2-to-1 advantage in men per point, the ""tall"" build has a lot of wiggle room for differences in castle distribution where it can still win.

A ""tall"" strategy will lose to a ""focused"" strategy that sends an unusual # of men to one or two castles (not enough to win by themselves) and then small #s of men to all remaining castles... but only if the ""focused"" player picks exactly the right castles. For example, a ""1-8-9-10"" tall player will lose to a focused-wide player that sends 54 men to castle 9 and 1 man-per-point to all other castles.

However, a ""focused"" build loses horribly to any ""wide"" build... and even to some ""tall"" builds. (for example, 9-focused versus 4-7-8-10 tall) Therefore, ""tall"" is the strongest overall strategy as it is only soft-countered by ""focused"".

When considering ""tall"" vs. ""tall"" fights... you're going to overlap on at least a few points. By definition if you win all of the overlap points, you'll have at least 28 victory points and you will win. So it is more important to contest the points you've chosen than to send single lonely soldiers to win uncontested points - you should go all-in on the limited # of castles you have.

Tall builds will be more likely to involve the higher numbers (8,9,10) than the lower numbers (you need all of castles 1-7 to win) so you should send greater-than-average men-per-point to the high numbers."
0,0,8,10,0,17,25,40,0,0,Targeted 28 points with what I guess is a relatively infrequently chosen combination and small number of castles.  Distribution of soldiers is a bit of guesswork on relative prioritization
0,0,8,9,10,14,27,32,0,0,"Just need 28 to win.  Tossed out 9 and 10 hopping to win the rest, need just 2 out of 3 from the 3:4:5 group.   Tried to put more on 8 and 7 to protect against 10:9:8:1 and 10:9:7:2 strategies."
0,0,7,9,11,12,13,8,20,20,Not a damn clue.
0,0,6,16,21,26,31,0,0,0,"(10+9+8) < (7+6+5+4+3+2+1), so the top three castles are negligible if I can win the bottom seven castles. Since winning the 6th castle once is the same as winning all three of the 1st, 2nd, and 3rd castles, it makes the most sense to load up troops in the middle four castles. Also assuming people naturally group numbers into multiples of five, I used a distribution of (multiples of 5)+1."
0,0,6,14,0,20,26,32,0,0,I chose what I thought the path of least resistance to 28 points was and put more on the higher ones.
0,0,6,11,18,27,38,0,0,0,I did (C-1)^2 + 2 for 2< C < 8
0,0,6,11,0,22,27,32,0,2,"I need 28 points to beat any opponent.  I figure most strategies out there will be of several forms: (1) get all the high point castles, so 10-9-8 plus something small; (2) skip the 10 and try to get something like 9-8-7-4 or 9-8-6-5; (3) get all the small castles, 7-6-5-4-3-2-1, and (4) some general ""what-is-each-castle-worth?"" strategy that has a declining point value for each castle.  To triangulate against them, I went with a very specific 8-7-6-4-3 strategy to try to get to exactly 28.  I also assume some human bias toward numbers ending in 0 or 5, so my numbers are 1 or 2 above those values.  Finally, I put 2 points in castle 10 to cover against those putting 0 or 1 in there.  Note that winning castle 10 would cover against losses three different ways: 8 or 7-3 or 6-4.  I don't expect to win, but I'm hoping that I'll place pretty high with this strategy, with an outside shot at winning."
0,0,6,10,1,22,28,31,1,1,"Attempt to get 28, and only 28 points."
0,0,6,8,10,11,13,15,17,20,"There are 55 points available (1+2+3...+10=55). 100 soldiers / 55 points = 1.81 soldiers per point. Starting at Castle 10 (most valuable) I send 1.81 x 10 = 18.1 soldiers, rounding up to 19 (I always round up to increase my chance of victory);  Castle 9 I send 1.81 x 9 = 16.29, rounded up to 17; and so on. I run out of soldiers on Castle 2, so I send that ""extra"" soldier to Castle 10, furthering my chances of victory there - conceding the 3 points available on Castles 1 and 2 for the 10 points available on Castle 10."
0,0,6,8,9,10,13,15,18,21,
0,0,6,0,0,19,22,25,28,0,Sunk cost
0,0,6,0,0,19,22,25,28,0,Sunk cost
0,0,6,0,0,19,22,25,28,0,Sunk cost
0,0,5,15,15,15,25,25,0,0,My son decided this was a good idea.
0,0,5,12,12,12,12,13,17,17,Experimentation in Excel.  Avg return was 37 points.  
0,0,5,10,10,15,20,40,0,0,"If I were playing this board game, that's how I'd start my first game."
0,0,5,10,10,5,10,15,20,25,
0,0,5,5,25,3,25,3,31,3,Because I am awesome
0,0,5,5,15,25,25,25,0,0,"I sacrifice the top two castles because that's where the bloodiest battles will be fought. I concentrate my troops  on 6, 7 and 8 where I make my best show. I will risk a few good men & women on the lesser castles in the hopes that one or two might be  a push-over."
0,0,5,5,0,15,20,25,30,0,"I choose to abandon the 10, instead focusing on the next best castles. Winning 6-9 is enough to win with even a little margin for error."
0,0,4,7,10,12,14,16,17,20,"I initially used an evolutionary model where I simulated the outcomes of various troop assignments excluding losing assignments in favor of winning ones. I then considered a value based model where each ith castle received (i/55)*100 troops so that 10 'value points' would be defended by the same number of troops whether they were all associated w/ one castle, or spread out among several. I used a monte carlo algorithm to benchmark both methods, testing each model against tens of thousands of randomized troop assignments. The value based model proved superior by a wide margin, but given how naive of a solution it was, I felt it wouldn't fare well against the majority of solutions. I noticed however that moving a single troop from the first castle to the tenth gave an assignment that beat this solution. This assignment however, could be outclassed in the same way. 

Thinking about the problem a little more, I realized only 28 points are needed to win, so I considered all or nothing strategies that heavily targeted just 4 castles with values adding up to 28. These solutions however, were the worst ones I'd come up w/ yet as losing just one of the 4 castles would guarantee failure against a typical assignment.

Intuitively, it was clear that 'clumping' was easily countered by more uniform troop distributions, but my benchmarking showed that the naive value-based model I came up with initially was worse than an augmented version of that model to a slight but statistically significant degree. Since the value based model was far better than the naive clumping approach, I decided to use that model as the basis for more monte carlo shenanigans. Starting w/ the value based model,
I generated random augmentations of a seed set of models . These augmented models differed by one to two troops assignments, and were benchmarked against randomized assignments like their predecessors. I would then collect the best models from one generation and use them as the seed set for the next generation. Eventually, I obtained a lot of models that look like the one I submitted, the submitted model being the best model in the last generation of assignments I produced. 

Looking back on my approach, I am optimistic about how this model would fare in competition. It looks a lot like the naive value based solution, but just a little bit better. I would hope my troop assignment would run into my value based solution a lot during the evaluation process, yet I have a few hangs ups. For one, I wonder if a clumping approach could work just by virtue of people playing fair. I imagine many people would submit well thought out solutions that might look a lot like my solution above. While my solution beats an estimated 95% of possible troop assignments, a clumping solution like [0,0,0,0,0,25,25,25,25,0] would beat my solution and others like it, while only beating an estimated 30% of solutions in general. Similarly, by benchmarking against random assignments, I may have included many extraneous strategies. Despite this, I don't feel fit to judge how motivated or clever other riddle enthusiasts are, so I'll stick to the result that took the longest to produce.
"
0,0,4,5,8,12,17,23,28,3,"I concede castle 10 in hopes that my opponent will wastefully over commit to win that castle. I also concede castles 1 and 2, because of their low value. Then I distribute my troops between castles 3-9 with an increasing emphasis on the later castles. Finally, I send three token soldiers to castle 10 in case my opponent was also conceding that castle for the same reason."
0,0,3,10,15,1,16,17,18,20,"I programmed a genetic algorithm to deploy troops. It came up with the idea of skipping castle 6.

Thank you for a really fun contest!"
0,0,3,4,8,16,0,5,32,32,"Simulation by estimating the distribution of what all the other players will do.
Obviously the issue is that I can't really know the other's distribution.
Unfortunately I didn't have time to have a model that distribute others players distributions into different broader modes :("
0,0,3,3,5,9,20,20,20,20,Trying to maximize the points against all possibilities.  I didn't use math - just a feeling.
0,0,2,26,2,2,22,22,22,2,To defeat the most opponents
0,0,2,16,3,20,3,21,4,31,"I selected castles (10, 8, 6, 4) as my win condition, committing 88 soldiers, with hedged placements on castles (9, 7, 5, 3) to contest easy point gain against plans with overlapping win conditions."
0,0,2,14,34,34,14,2,0,0,"I assumed most people would either play to the low end or the high end (e.g. A bimodal distribution of allocated points). I countered with a simplistic and crude normal distribution.  I could have easily assumed people would play to the mean and avoid the high/low castles.  However, I only submitted one strategy and a coin flip chose this one. "
0,0,2,12,15,15,16,20,20,0,Scarifice low and high level bases but try to win 4-9.  4-9 = 39 of 55 total points.  6 bases (60% of bases number but 71% of value).  Plus game theory for level 10 where everyone else will go there I try to zig when they zag.
0,0,2,3,5,8,12,17,23,30,
0,0,1,13,1,1,23,27,29,5,"Distribute relatively proportionally by value of castle 92 troops across 9, 8, 7 & 4 (adds up to 28 - the minimum needed to win) and then distribute 8 troops relatively proportionally across 10, 6, 5 & 3."
0,0,1,11,12,14,41,21,0,0,"There are 55 points available so I need to get 28 to win. If i take a top down approach i need to take castles 10,9,8 and 7. A bottom up i need to take all castles from 1-7. middle of the road Castles 8-5 and either 2 or 3 or 4. 7 is the most important in stopping either a top down or bottom up wins so this is the one most needed followed by 8 and 6. This method will beat simple approaches like 10 troops on all or 20 troops on the top 5 but may not work against any players opting to take 5 and 6 decisively.
SUBMIT
"
0,0,1,11,11,16,26,31,2,2,"I saw that taking castles 8, 9, and 10 would give 27 of 55 points and an almost certain victory, so I knew that if many people opted for the more valuable castles, 8 was the number I could not give up. I allocated 95 of my soldiers to castles 4-8, because giving up 9 and 10 meant that the bottom three were the most I could forget about. I made the number of soldiers at these five castles one greater than a multiple of five, because I thought that some people would choose multiples of five. I made sure to leave some soldiers over for the top two castles, so I could take them if my opponent left them alone. The final soldier went to castle 3, in hope that I might get those points as well."
0,0,1,6,14,19,1,20,28,11,"I wrote a (probably unreliable) genetic algorithm to test many different strategies and evolve an optimal one. It suggested that the ideal strategy was to aim for a coalition of castles 5, 6, 8, and 9. I am using its ""best"" result."
0,0,1,3,3,5,20,21,45,2,Giving up the 10 and the least valuable castles in order to have a better chance at winning the others which cumulatively are worth more.
0,0,1,2,3,11,13,18,24,28,"I wanted to build a strategy that would be strong against balanced attacks and  have enough depth if someone wants to dominate the high value castles. I avoided numbers that end in 5 or 0 to prevent ties, rather picking the number above the midway. (13 as my 7 castle is above the midway point of 12.5.)"
0,0,1,2,3,6,13,25,50,0,
0,0,1,1,21,21,2,26,26,2,"Banking on winning castles 5, 6, 8, and 9. Set winning values for 8 and 9 assuming that someone did an equal distro of 25 soldiers to 4 different castles. Placed 2 soldiers at 10 and 7 to account for people who placed individuals at various castles and large volumes at other castles."
0,0,1,1,1,20,25,25,27,0,"A very slight improvement over my previous attempt. If I can only submit one deployment, feel free to remove my other two deployments.

I realized that my idea (dominate four castles) would also be found by others. So I'm making a deployment that beats that one... slightly. 

(As you can tell, I'm a little fixated on this game.)

Thank you so much for creating and adjudicating this game!
"
0,0,1,1,1,1,20,35,40,1,"I figured I'd give up 10 mostly, and focused on 987, which are worth a lot but hopefully not as contested"
0,0,1,1,1,1,19,25,26,26,"The strategy is not proposed based on its own merits, but rather the fact that it counters so many other deployments. The 25 across all 4 top castles strategy loses to this one because you guarantee to beat them in at least 2 plus the lower echelons. In fact I can see no plans that beat this one."
0,0,1,1,1,1,1,18,19,58,I'll explain if I win.
0,0,0,33,33,34,0,0,0,0,Figuring have a better chance to win more of the lower ones as long as most people go for the larger ones. 
0,0,0,33,33,22,3,3,3,3,Most people will start high to low is my guess so I put 0s there to not compete where I can't win. A % of people will not compete at all for the 7 to 10 spots if they load up toward other castles early on. And the 3-6 spots should include less of peoples weights - my play is less of a stats based play and more of a sociology play
0,0,0,25,25,25,25,0,0,0,"I figured there would be lots of different strategies tried on this problem, thus the first couple of obvious ones (send 100 troops to castle 10, send 10 troops to each one of the castles) would likely be used.  For some reason (call it a hunch on human nature) I figure there will be a more likely event that people will place a number of troops either in castles 1-3 or 8-10, but I feel like the odds are lower that they will place troops in the middle 4 castles 4-7.  I also felt like there would be a better shot at winning castles if I ""didn't get greedy"" and go for either a large number of castles, or castles with the highest point value.  So, here goes nothing!"
0,0,0,25,0,0,25,25,25,0,4+9+7+8=28 which is the least amount needed to win the war.
0,0,0,25,0,0,24,25,25,0,"10*11/2 = 55 points in the game, score of 28 wins,  need at least 4 castles, so 4 castles it is, avoid the bing guns (e.g the 10+9+8+1 combo) since vegas addicted people will front-load the 10,  the only other 4 castles combo without repeat is 9+8+7+4,  even spread of ressources between castles since they are all equally important in this strategy. "
0,0,0,24,0,0,25,25,26,0,I tried to ensure victory at the minimum number of castles that would give me the minimum number of points to win.
0,0,0,21,0,0,26,26,27,0,"There are 55 victory points, so we need to win 28 to win the war.  No three castles provide enough victory points by themselves, so my strategy is to concentrate on four castles that are cumulatively just enough to win the war, avoiding the most attractive castle #10, and concentrate all my troops there to maximize odds of winning these castles. Castles 4, 7, 8, and 9 work.  (Another similar choice would have been castles 5, 6, 8, and 9.)  I distributed the 100 armies across these castles with slightly more on the more valuable castles rather than evenly."
0,0,0,20,20,20,20,20,0,0,complete guess 
0,0,0,20,20,20,20,20,0,0,Submission #2. I'm not convinced this will be successful either. I abandoned the low AND high end value castles in favor of equally dividing them amongst the middle tier of castles (that would still give me a winning score).
0,0,0,20,20,20,20,20,0,0,"I believe the middle group of castles (4-8) might be undervalued relative to the high end. Since I need 28 points to win, I want to make a very concentrated win those points in the most likely way I believe is possible. In this way I would win 5 castles but more than half the points. Since I need all 5 castles to win, I deployed the soldiers evenly."
0,0,0,20,20,20,20,20,0,0,Go for the ones that are in the middle so you get the ones with the least intrest.
0,0,0,18,19,20,21,22,0,0,"Let him over-allocate for the two big prizes, use just enough to win the 4-8 slots"
0,0,0,18,19,20,21,22,0,0,I focus on 5 middle castles that allow me to just win more than half the points
0,0,0,17,0,0,17,33,33,0,Minimize castles.  9 & 8 are most important.  10 will probably be overvalued.
0,0,0,16,1,1,21,26,35,0,Goal to capture 7 thru 9 (because 10 will be contested) at which point I only need 4 more for 28
0,0,0,15,25,30,30,0,0,0,Collect mid-range castle with high concentration of troops.
0,0,0,15,20,21,21,23,0,0,Guesswork
0,0,0,15,18,0,0,0,32,35,"Used the minimum number of castles to get to 28 points, and then allocated for the highest average win probability.    "
0,0,0,15,15,20,20,30,0,0,"Sacrificed 10, 9, 3, 2, 1 because that bumps the avg troops per castle by 10. Then did an even dispersement from highest to lowest. If no one knows that you're taking this strategy it gives you an advantage as they are likely to put the largest portion of their forces into 10 & 9 leaving a minority portion to take on your entire force. 8, 7, 6, 5, 4 will earn the win and gives you a little breathing room should you have to split any."
0,0,0,15,15,0,0,0,35,35,"There are 55 points up for grab, so any strategy should aim to win at least 23 victory points. Castle 10, 9 and 8 contain over half of the victory points in the war, so to get a majority of the points any strategy must attempt to win at least one of these. Also the minimum number of castles you need to get 23 points is 3 so you shouldn't waste troops fighting over a large amount of castles but focus on winning a few key ones.
After getting that far in my thought process I couldn't decide what I should do, so I wrote a simple simulation to find interesting strategies for me. It randomly generates several thousands strategies and makes them fight in the same way you will be judging the contest. After that they are ranked by how many victories they achieved and then the losing half is removed and replaced by new strategies generated by randomly modifying ones in the winning half. This causes strategies to evolve over time. Strategies would rise up and start dominating the simulation then eventually be bested and move down to the losing half and disappear.
The strategy I chose was one that I hadn't though of but my simulation did, so I'm hoping not many other people will have thought of it. It dominated the simulation for a decent amount of time when I didn't expect it to. It focuses on winning castles 9 and 10 and hedges its bets between 4 and 5. As long as I win 9 and 10 I only need 4 more points to get to 23, which lets me focus on castles that might not be very hotly contested."
0,0,0,15,0,0,26,28,31,0,"I need 28 points to win, so I'm trying to win castles 4, 7, 8, 9. Each of those I placed roughly n/28 * 100 troops. I'm hoping anyone using a strategy other than this would not place so many troops on any of those four castles. "
0,0,0,15,0,0,25,30,30,0,"I know I have to win 28 value points to win the war.  I think many people will overvalue Castle 10, so I'm punting.  I then have to load up in the hopes I can blow away Castles 7-9 with the bonus troops I saved... that puts me at 28 pts.  I then need only 4 more... I was split between trying to win both 2 & 3, or just going for 4, but I'm consolidating everything & seeing how it pans out. #trusttheprocess"
0,0,0,15,0,0,25,28,32,0,"I considered a handful of common strategies and counter-strategies, and then picked castles that added up to 28 points (4,7,8,9). I assigned soldiers to each castle so that the ratio of soldiers to points was about constant- close to 100/28, noting that people who go for castles 1-7 might assign 14 to 4. "
0,0,0,15,0,0,25,25,35,0,Concentration of force on the lest amount of numbers needed to win greater than 50% of the victory points while avoiding the highest value numbers if possible.
0,0,0,15,0,0,23,31,31,0,"Only need 28 points. Fewest number of castles is 4. Wanted to avoid castle 10, which is often going to cost the most to win, so I'm going for castles 4,7,8,9. Few people will contest 4 with more than 14. Few people going for the 1, 8, 9, 10 win will put more than 31 on 8 and 9. I'm vulnerable to people who put 33 on 8, 9, 10 and only 1 on castle 1, but they're going to be vulnerable to all randomly chosen strategies and strategies that go for Castles 1-7 (strategies I'm robust to). "
0,0,0,15,0,0,20,30,35,0,"I decided to consolidate my troops. As long as I win each of the castles I have sent troops to, I will have a majority of the victory points. I think I will rarely meet an opponent more invested in my castles than I am."
0,0,0,14,18,0,0,0,33,35,"55 total points divided by 100 soldiers gives each soldier an avg. point-to-soldier value of .55, and vice versa 100 soldiers divided by 55 points gives an avg.  solider-to-point value of 1.81.  28 points is the minimum point to win, so any soldiers expended to guard castles beyond that point total are non-optimized.  Multiplying each castle's value by 1.81 yields the average soldiers required to garrison that castle (ie, Castle 1~ 1.81, Castle 2~ 3.62, Castle 3~ 5.43, etc.).  When rounding to the nearest whole number, Castles 4, 5, 9, and 10 are the least expensive to garrison (have the smallest soldier-to-points ratios), and they also happen to add up to 28 points.  I sent double the minimum soldiers to each of those four castle in the hopes that, on average, this strategy will optimize on the best points-to-soldier ratio (ie, a point-to-soldier ratio of approx. 3.6)."
0,0,0,14,18,0,0,0,32,36,"Select castles [4,5,9,10] and send troops there in proportion to the victory points in each."
0,0,0,14,0,21,0,29,0,36,"There is no unbeatable solution. Capturing 10,8,6,and 4 wins. So concentrate there proportionally and hope for the best. "
0,0,0,14,0,21,0,29,0,36,Point value divided by 28 points to win.  Multiply that percentage by 100.  Choose fewest number of castles to get to exactly 28 points.
0,0,0,14,0,0,25,29,32,0,"With 55 total points available, I only need 28 points to win.  The least amount of castles I need to get 28 points is 4. There are 9 ways to get 28 points exactly with 4 castles. i chose one of the ways that doesn't use castle ten."
0,0,0,14,0,0,25,29,32,0,minimum number of castles to win(4) and minimum values of castles (9+8+7+4=28) then allocate troops proportionally. Avoiding 10 because ppl will be tempted to take it.
0,0,0,14,0,0,25,29,32,0,"One only needs 28 points to win the war. The fewest castles one needs to win is four. There are only two configurations of four castles that add up 28 points, 10, 9, 8, 1 and 9, 8, 7, 4. I chose the second group because I can lose the 10 point castle and still win. I weighted the armies to the castle by what percent the points earned are out of 28. "
0,0,0,14,0,0,25,28,32,1,"With 55 total points, 28 are needed to win.  The 10 point castle will likely be highly contested so I ignored it.  The 1-5 point castles aren't enough points for me to focus on.  But if I can seize the 6, 7, 8, and 9 points castles I'll have 30 and the win.  I need to get all of them in order for this to work though.  

Just in case anyone else adopts my strategy, I am dumping the 6 castle and pursuing the 4 castle partially because I am a pirate and like math/ship puns but mostly to throw anyone else off my trail.  This puts me at 28 points.  

I weighted each castle based on the percent of points gained (4/28, 7/28, 8/28, and 9/28) to get a percent of my 100 soldiers/pirates to deploy.  Rounding down, I have 14 pursuing castle 4, 25 pursuing castle 7, 28 pursuing castle 8, and 32 pursuing castle 9, with 1 left over from rounding.  I put the extra one toward castle 10 in case my opponent also left it alone."
0,0,0,14,0,0,18,29,39,0,"Fewest # castles, estimated e^0.5x population distribution."
0,0,0,13,16,22,23,26,0,0,"After realizing we didn't have the skill to enumerate all strategies we decided to iterate strategies based on whether they were able to beat an agreed default strategy - 10 soldiers in each castle. After playing with this iteration on paper for a bit we found this strategy was the strongest we came up with. It is certainly not an equilibrium strategy because it loses to 33 soldiers in castles 8-10, with excess on castle 1. It would be interesting to receive results even though I'm certain this will not be the winning strategy."
0,0,0,13,16,20,24,27,0,0,I focused on castles that would give me 30 points (more than half available).  I ignored castles 9 and 10 thinking they would be the most desirable.
0,0,0,13,16,19,23,27,1,1,"Picked something that added up to more than half the points, and subdivided my soldiers proportionally among them. I skewed slightly high in hopes of defeating others that stacked too heavily on top of 10."
0,0,0,13,14,17,16,19,21,0,I did the opposite of Egyptains in 1968
0,0,0,13,1,1,21,26,37,1,"I chose ""concentration of forces"".
4+7+8+9=28, which is enough to win. I only have to get these 4.
For each one i alloted enough sodiers that it is unlikely the oppositon has more.
Just as a backup against a similar lopsided Opponent, i gave 5,6,10 just 1. "
0,0,0,13,0,0,24,29,34,0,Concentrate all of your forces in the minimum set of numbers required to get to 28 points (aka electoral college strategy!).
0,0,0,12,19,0,21,4,21,23,A genetic algorithm told me to.
0,0,0,12,16,20,24,28,0,0,"I focused on only castles 4-8. If I win 7 and 8, I'll be down 8 (losing 1,2,3,9,10). So I need to win 2 of 4,5,6. Or win one and tie the other two. "
0,0,0,12,2,2,21,31,31,1,"Focus on winning castles 7-9, with hopes to scrap up the remaining 4 points at castle 4. I tried to ensure that I'd beat what I viewed as the most basic strategies, flat even (10-10-...-10), I'm-smarter-than-you even (1-11-11...11-11), and top down (0-...0-10-20-30-40). It loses to I'm-even-smarter-than-you even (2-2-12-...12), but hopefully that's an acceptable loss."
0,0,0,12,0,0,25,30,33,0,"Need 28 points to win so just focusing on putting troops at least amount of castles to get to 28, so 4 castles.     Avoided 10 because may be most heavily guarded at other castles so only other combo is 9,8,7,4.   Weighted 9 and 8 higher, 7 slightly lower, 4 lower than that.  "
0,0,0,12,0,0,24,29,34,1,Trying to win 28 points exactly. One troop reallocated to castle 10 in case I get unlucky and lose castle 4 then maybe I can pick up 5 or 10 from winning or splitting castle 10.
0,0,0,12,0,0,20,34,34,0,"Wanted to focus my efforts on the easiest way to get to 28, but not focusing on 10 (which will probably have most of the opponent's soldiers).  Taking a risk to let the other 6 castles fall."
0,0,0,12,0,0,20,30,38,0,Don't waste any soldiers. Get to 27.5 points. 
0,0,0,12,0,0,16,34,38,0,"I chose to focus on 4 castles, because that is the fewest number of castles that would provide a majority of the 55 points available. I figured that the most obvious way to focus on 4 castles would be to go for 10,9,8,1; the alternative, which I expected fewer people to focus on, is 9,8,7,4. I placed the bulk on 9 & 8, seeing those as the most important, in order to defeat someone who was focusing on the 10,9,8,1 strategy. I then left just enough on both 7 and 4 to defend against anyone who happened to do an even 10 distribution on all castles, as well as anyone who did a variation of a proportional distribution (~1.8 soldiers per victory point)."
0,0,0,11,21,0,0,0,31,37,"The total number of points available is 55. So, in order to win, I must get at least 28 points. Rather than spreading my troops out, I decided to put all my eggs in one basket and attempt to receive the minimum number of points. I figured that most people would stay away from numbers 10 and 9, since those are too obvious. Taking the contrarian approach, I decided to invest heavily in these two castles.  This gives me 19 points, so I must add a further 9 points to my total. The easiest way to do this was simply choose castles 4 and 5 (I wanted to stay away from castles 6, 7, 8 as I figured these would be very popular.) I invested my remaining troops there. This gives me a total of 28 points, just enough to win. Additionally, when assigning troop values, I tried to have the ones digit of my troops 1 and 6, so I would beat simple multiples of fives.  "
0,0,0,11,13,17,21,38,0,0,"Submission #6. I have the most faith in this, my final submission. Since I have little faith in this one as well, however, I think I would properly be considered a ""pessimistic general"".  Anyway, I chose to only defend the middle value castles and to divide my forces unevenly rather than equally. I know there are lots of other options than the three choices (with two variations each) that I chose but they seem like a good set of choices to me!  Thanks for the fun riddler!? (I'd love to know how many submissions you received and where each of mine fell in the rankings but I'm sure that's WAY more work than you're prepared to put into this. Take care!"
0,0,0,11,12,13,14,15,17,18,"Tried to weight the higher point castles more, but distribute troops.  Ignored the low value castles."
0,0,0,11,11,0,26,26,26,0,I thought about strategies I thought large number of people would pick and chose a strategy that would perform well against those strategies.
0,0,0,11,2,2,17,34,34,0,"Surrender the 10er, but get 7, 8, 9 and 4 for 28 - enough to win. I need 11 on 4 to beat a 10 x 10 strategy; I need 34 on 8 and 9 to beat a 34-33-33 strategy. I initially had 21 on 7, but I hedged my bets just in case I run across a copycat strategy. (7 on 21 loses to a 25-25-25-25 strategy anyway, for example!) 

I could very very well fall in the first round, but I can also see this strategy work well. "
0,0,0,11,1,1,24,26,36,1,"Hopefully the opponent sends several troops to 10. By sacrificing 10, I can concentrate on 7,8, and 9. Winning those will put me 4 points away from clinching. I also sent 1 to several castles I was going to leave blank just in case my opponent was also going to send 0. Then I get a very cheap win at those castles."
0,0,0,11,0,0,29,29,31,0,"I looked at different ways to get 28 points and thought that committing to 3 big numbers hard and one small number that makes the total 28 could be a good strategy. I debated between 9,8,7,4 and 10,9,8,1 and in the end decided skipping out on the 10 fight is probably worthwhile. 11 on the 4 beats the 10 across the board plan and the 29 on 7 beats the even split across the lower 7 numbers plan."
0,0,0,11,0,0,26,31,32,0,"There are 55 pts to get, any path to 28+ requires at minimum 4 castles (10+9+8 is still not enough). I am letting go of #10, which I expect to waste a lot of soldiers without being more useful in the 4-castles path to victory. All desired castles should have more than 10, to beat the equal spread in every castle. I expect big castles to often go for over 25 (4*25 path to victory). However, I am betting I can get my fourth castle (#4) much cheaper (between 11-25, thus saving soldiers to secure 9+8+7). I expect to lose some matchups (especially ties), but to do well overall. But then again, strategical thinking against an unknown crowd is harder than tactical games or strategical games against known opponents. If everybody sees the 4 castles path and nobody goes for wide spread, I might be crushed every time since I have no back-up."
0,0,0,11,0,0,26,30,33,0,"Put one extra soldier (than allocation by normal victory point value) at castles 7, 8, and 9, then put remainder at castle 4, so that if all 4 castles won, the total would be 28 out of 55 points.  Hopefully, I can win all castles 7, 8,and 9 much of the time, and castle 4 also some of the time."
0,0,0,11,0,0,26,30,33,0,"As per my e-mail to Oliver, I have previously submitted an entry but I would like to change it to the above.

My thinking has not changed all that much. I still think that a proportionate strategy is best but that it is also a good idea to aim for enough points to win, rather than all of the points. I also think that a strategy needs to be robust against similar strategies and to defeat all obvious strategies.

My concern, however, is that an opponent who anticipates the popularity of a proportionate strategy could exploit this by making their strategy slightly less proportionate, for example by overloading castle 10 at the expense of one of the other castles. This would defeat a proportionate strategy which targeted the same castles.

The best way to counter this is to not target castle 10 at all, and to let those smart alecs waste their resources.

Therefore, my strategy is now an almost-proportionate 28 point strategy, focusing on castles 4, 7, 8, 9. It is only non-proportionate to the extent that it takes 3 troops away from castle 4, and adds one to each of castles 7, 8 and 9.

I have done no computer simulations - this is all in my head - but as far as I can see, it will do very well against almost every opponent.

It will defeat every fully proportionate strategy and every non-proportionate strategy in which the emphasis is on overloading castle 10.

It will also defeat a flat strategy, in which 10 troops are allocated to each castle, or 25 troops are allocated to each of four different castles."
0,0,0,11,0,0,23,28,33,0,"It's impossible to win with only three castles -- even if we take the best three castles we'll have fewer than half the points. But it's possible to win with only four castles.

The optimal strategy is one which guarantees 28 or more points as often as possible. With the four castle strategy, we must win Castle #9 or #10 or we can't reach 28 points. I chose Castle #9 which would be less contested than #10. This surprisingly leaves only two options: [9, 8, 7, 4] or [9, 8, 6, 5].

Between the two, [9, 8, 7, 4] just seemed a little more consistent to me. I think a reasonable player might commit 15 or more soldiers to Castles #5 and #6, so it might be more  of a guessing game. But 11 soldiers will almost certainly win Castle #4."
0,0,0,11,0,0,22,22,22,23,"There are 55 victory points available, so I need 28 to win. The smallest number of castles I can do this with are Castles 10, 9, 8 and any one of the others. In order to have a margin of error, I decide to target Castle 7 additionally and out of the ""any ones"", I target Castle 4. This way, I need three out of the four large ones plus Castle 4. I reckon Castle 4 will not be targeted so often, so I only go with 11 soldiers there, allowing me to beat an ""evenly distributed"" strategy. The large castles get 22 soldiers, beating any ""target only the largest five evenly"" strategy and even those that assign 21 soldiers to beat these. That leaves one odd soldier, who I send to the largest castle. "
0,0,0,10,20,20,50,0,0,0,Trying to win the mid castles
0,0,0,10,20,20,25,25,0,0,Mostly Random with emphasis on numbers between 5-8.  
0,0,0,10,20,20,25,25,0,0,"I only need 28 points to win, so I'll focus on winning the less popular castles."
0,0,0,10,16,17,18,19,20,0,"Completely ignore the top point getter, thinking that every one else would send lots of troops there. I concentrate on the next best, and decreasing down the list until I run out of troops."
0,0,0,10,15,20,25,30,0,0,"My assumption is most people would send a majority of their troops to the largest castles, 10 and 9. By starting for castle 8 with a modest sized team and incrementally decreasing the amount, I could collect 30 victory points without vying for the biggest prizes."
0,0,0,10,15,20,25,30,0,0,"Well, I think most people would try to stack their troops in the higher numbers. But if I win middle from 4-8, then it would give me a higher total."
0,0,0,10,15,20,25,30,0,0,"*There are 55 total points, so we are trying to figure out the path to 28. How can we do this in the smartest way? 
*Let's think about where we can concede points and hopefully make our archenemy overpay. 
*In strategy games like this, the extremes tend to draw the majority of the attention, so we will immediately concede 1, 2, 3, 9 and 10.
*This leaves me with 100 soldiers to allocate towards 4-8.
*Assuming the archenemy has some sort of distribution weighted towards the more valuable castles (unfortunate we do not know anything more about their mindset!), we will need to allocate more soldiers towards 8 than 7 and more towards 7 than 6, etc.

*Side note* I could have cheated a bit here, because the question at the end says ""Whoever wins the most wars wins the battle royale and is crowned king or queen of Riddler Nation!"" so logically I could just aim for winning Castles 1-6 assuming each of those is a ""war"" and disregarded the ""point system"" =)"
0,0,0,10,15,20,25,30,0,0,"Only need 28 points to win and high point castles are most valuable, but don't compete for the highest value castles; they are likely too expensive to win. "
0,0,0,10,15,20,25,30,0,0,"The lowest castles arent worth enough points to commit troops, and the highest castles are likely to be overvalued by my enemies. If i win these middle castles my 30 points will win the war."
0,0,0,10,15,20,25,30,0,0,Ignore the top and bottom focus on the middle
0,0,0,10,13,20,26,31,0,0,I assume most will attempt to aim for the top only. I'll take the center-top instead.
0,0,0,10,10,20,20,40,0,0,Everyone is going to put lots of soldiers on the top castles.  So I give them the 19 points hoping to get a lot of points from the five I defend.
0,0,0,10,10,10,20,30,0,0,"Didn't want to compete in upper end, or lower end, and figured had a better shot at winning the middle"
0,0,0,10,10,10,10,15,20,25,Take the positions that earn the most - abandon the weak ones. 
0,0,0,10,0,21,0,21,0,48,"Bet on fewer castles and ignore the one immediately below it, keeping in mind to bet on enough castles to get over half the available points."
0,0,0,10,0,0,30,30,30,0,"If I usually overwhelm the opponents on 4, 7, 8, and 9, I'll get 28 points, which is enough to win the war."
0,0,0,10,0,0,25,30,35,0,"I expect 2 general deployments:
1) Equal deployment, or downmarket. Trying to win as many castles as possible. These folks will have either equal, or concentration on the lower castles. I beat them at the higher castles, and tie on at least one of 4 and 10.
2) Upward mobility. These folks gamble on capturing 8, 9 and 10. 10 would then have the highest concentration. So, I hope to win most of 7, 8, 9 and 4, plus tie on any other lower castles that they left empty."
0,0,0,10,0,0,20,34,35,1,"The maximum possible points is 55, so more than 27.5 are required to win. So the strategy should be to try to get enough castles to guarantee at least that many points, and not waste troops on trying to get more. There are of course a variety of ways to do this - e.g. holiding castles 1 through 7, or 4 through 8, or 19, 9, 8 plus at least one other one. Since at least 4 castles are required, I figured I would go for a strategy that requires 4 but NOT castle #10, which I imagine people on average will try to defend heavily. So I opted for 9, 8, 7, and 4. I am putting 1 person in castle 10 so that I can win a tie if someone else uses a strategy that doesn't involve castle 10 and happens to win over one of my other castles. I figure that even in the 8-9-10 concentration strategy, people are unlikely to use more than one third of their troops in a castle, particularly those less than 10, which is why I stuff 35 and 34 in 9 * 8. 7 is probably most vunerable, but if someone is using a strategy that involves castle 7 then they are likely diluting troops and chances are they have less than that there. Castle 4 is essentially a random small castle, so if someone is diluting troops to put there, I figure 10 (100 total/10 castles) is a fair number here. "
0,0,0,10,0,0,20,30,40,0,I figure most people will want the 10 points at castle 10 and will be willing to spend close to half their budget on it. My strategy is to get exactly the number of points needed to win (28) so I'm going to load up on the minimum amount of battles that I need to win in order to get those points. Sending no one to 10 ensures I'll have more to use against my opponent at the next three battles. 
0,0,0,9,14,19,24,34,0,0,Using the lowest possible combination to achieve over 28 points and deploying troops proportionally to the value of the points available.
0,0,0,9,11,12,14,16,18,20,"Total number of points available is 55.  100 soldiers / 55 points = 1.81
I picked:
1.81 * pt value of castle + 1 (rounded up)  for the 7 most valuable castles
0 for the 3 castles with the least points."
0,0,0,9,0,0,21,27,43,0,"It does very well against uniformly random and proportional deployments.  Additionally, by ceding the battle at 10 I hope to avoid a lot of wasted troops while maintaining a winning coalition."
0,0,0,8,10,12,14,16,19,21,"Cede castles #1-3; assign each remaining troop in proportion to the value, giving an extra troop to castles 9 & 10. It makes the numbers work nicely."
0,0,0,7,9,11,14,17,20,22,"In total, 55 points are up for grabs. 100 troops means that you have roughly 1.8 troops per available point. A simple solution would be to just multiply the number of points per castle by 1.8. However, since each castle is winner take all, alotting an extra point to a high value castle is worth more than removing a point from a low rank castle. I therefore gave high rank castles somewhat more weight. Very low rank castles are not worth defending at all if it substantially decreases your chances of winning higher rank castles. The exact balance is based on intuition rather than calculation."
0,0,0,6,2,2,26,31,31,2,"I need 28 points to win the battle, and so taking the 4, 7, 8, and 9 point castles will achieve that. It is better to spread troops over fewer castles so I choose to concentrate on these four. Hopefully boosting the numbers in the 9 and 8 spaces will counteract someone vying for a (10, 9, 5, 4) or (9,8,6,5) win.  These are must haves for me.  I drop a few troops in the 10, 6, and 5 slots in case my opponent hedges a single troop there.  This means if I lose the 7 I may be able to make up the points if my opponent is negligent. Being somewhat negligent myself, I ignore the final three castles."
0,0,0,6,0,0,26,34,34,0,"Tried to find a way to 28 points that was different than my first, simpler idea. Chose 34 for castles 8 and 9 thinking a strategy to win 8, 9 and 10 could use 33 for those. "
0,0,0,5,15,0,25,25,30,0,"28 points is needed to win.  I'm expecting most people to put troops into castle 10, since it's worth the most points, so I'm letting them waste them there.  I'm focusing my troops into castles 7, 8, and 9, since together they will net me 24 points.  To get the last 4, I'm putting most of my remaining troops into castle 5, with a backup plan for getting castle 4 instead.
I consider this strategy an all-or-nothing gambit; if I don't win castles 7, 8, and 9, I'm pretty much out of luck"
0,0,0,5,15,0,20,20,40,0,Just need to get to 28
0,0,0,5,10,10,15,30,30,0,I tried not to waste troops on places with no reward (castle1) or somewhere my opponent might make it hard to win (castle 10)
0,0,0,5,0,0,15,0,50,30, ??\_(??)_/??
0,0,0,4,10,11,15,30,30,0,Mostly just guesses. Math is hard.
0,0,0,4,9,12,15,18,21,21,"Assumed descending from highest to lowest, then ran a boatload of simulations against similar strategies. Randomized a tad just because."
0,0,0,3,13,21,21,21,21,0,Trying to metagame a bit - going for 4-5 numbers exclusively seems optimal but this gives me the ability to beat a number of strategies (like 20-20-20-20-20 in any 5 numbers).
0,0,0,2,19,23,27,29,0,0,Get to 28!
0,0,0,2,5,10,17,24,20,22,https://github.com/schwar569/riddler_nation
0,0,0,2,2,22,23,24,25,2,
0,0,0,2,2,2,30,30,32,2,Get 9/7/8 and then hope the 2 on some other castle is enough
0,0,0,2,0,0,28,34,34,2,"28 out of 55 points are required to win (it's like 270 out of 538!).  Since 8+9+10 = 27, no 3 castles will guarantee a win.  So, rather that contest the 3 most valuable castles, I chose to try to assure winning castles 7, 8, and 9 for a total of 24 points.  Then I need to win 4 more points.  I chose to do this by making a play to win castle 4 or win or split castle 10.  I chose 34 soldiers for castles 8 and 9 in case my opponent was dividing his soldiers nearly equally among 8, 9, and 10.  I chose 28 soldiers for castle 7 in case my opponent was trying to win castles 1-7 for 28 points and divided his soldiers nearly equally among these castles. By the way, my first thought was to try to formulate a strategy that would ensure at least a tie, but computer experiments with a few castles suggested that this was not possible."
0,0,0,1,11,26,0,31,31,0,Aiming to balance troops in just enough castles to get to win the war.  Trying to balance it to beat evenly rounded numbers other people may use.
0,0,0,1,5,6,16,19,26,27,"Two things to note before going to the computation method. Firstly, the deployment 10-tuple should be weakly increasing; it makes no sense to devote more men to a less valuable castle. Secondly, there is no single deployment that beats all others. To see this, observe that (0, ..., 0, 100) is not best, as it is beaten by (0, ..., 0, 50, 50, 0). For any deployment other than (0, ..., 0, 100), take the first castle to be assigned a number of soldiers >0, and move one of them to the next (higher value) castle. This slightly modified deployment will win. [As the set of possible deployments is finite and ""beats or ties"" is 'total', we can see that ""beats or ties"" is not an ordering relation.]

This means that we're not trying to find (or approximate) the 'top' deployment, as there isn't one. Rather, we're trying to find the deployment that will beat the majority of submissions. As a first guess, let's randomly generate a bunch of (weakly increasing) deployments [more on that below], and see which ones beat the most of the other ones we generated. We might be tempted to just see which one gets the highest winning percentage and put that one in. But, we're not trying to get the highest winning percentage against random deployments, but rather against submitted deployments. So, we want to see which deployments work best against ""good candidates"" (assuming that's what people will submit).

We do this iteratively. Generate a whole bunch. Send the top half through to the next round. Then play that top half against each other, and send the top half of that round's competition through to the next round after that (irregardless of how well they did in the previous round; like Masterchef, unlike Hell's Kitchen, for any fellow Gordon Ramsey enthusiasts out there). At each iteration we get a better version of ""see how they do against 'good' candidates.""

My submission was the best after 10 rounds of this. I'm sure more rounds would give a better submission, but we'll see how this does.

Finally, the messy details about how I made random weakly increasing deployments.  I generated 10 random independent samplings of the Unif(0,1) distribution. I then scaled them all so as they summed to 100, rounded each of them down to the nearest integer, and added whatever I needed to the last sample to make them sum to 100.  Then, I sorted them, assigning the lowest integer to the least valuable castle, etc."
0,0,0,1,1,1,12,12,33,40,"Base strategy of 37,33,11,11,1 for castles 10 through 6 with 7 spare soldiers for flexible deployment. Then into thinking about what others would play to 'optimise' final distribution."
0,0,0,1,1,1,2,1,3,1,"I took a computational approach. First, generate a population of random strategies, then pit them against each other. Save the top 50% of the strategies by win % and propagate them to the next round. Repeat for 10,000 rounds.
An interesting property of this game is that strategies are non-transitive in head-to-head matches. Additionally, the best strategy depends on the distribution of the strategies in the competition pool. The computational approach that I took assumes a competition pool of 1/2 ""good-ish"" strategies and 1/2 random strategies. Here's hopin'! Code at https://github.com/cjbayesian/riddlerfivethirtyeight/blob/master/Riddler%20Classic%20Battle%20Royale.ipynb"
0,0,0,0,50,50,0,0,0,0,"Figured someone would send 100 to 10, so the easiest way to get to 11 castles was splitting between 5 and 6."
0,0,0,0,25,25,25,0,0,25,"You only need 28 points to win - and intuitively any troop you spend on a castle you lose is wasted, as is any troop you spend on points beyond your 28th point. So is any troop you spend on a particular castle you win beyond what you needed to beat your opponent. Targeting castles 5,6,7,10 adds up to exactly 28 points if you win all 4, so if you can win those 4 without overspending on any of them you played well. I played with other combinations of castles but in simulations they didn't win as frequently. I also looked at other ratios of castles but an even split won out in the end. In a bloody melee with 100000 randomly chosen bots playing against every other bot (using a few different strategies to choose troop allocation but mostly dice-rolling), this simple approach won 97.9% of the time - the highest rate. I then filtered those bots to only those that won 80+% of the time (3905 bots reached that threshold), and re-ran the competition. Playing against this high-talent pool (surely the closest analog to the Riddler contestants...), this approach still came out on top, winning 98.7% of the time."
0,0,0,0,25,25,0,25,25,0,"Note that at least 4 castles are needed to win. In general, I'd expect people to place more troops at higher-value castles. I've gone for 4 castles which have enough total value to win, but which should hopefully have be the easiest to win (I expect people to put most troops at higher value castles, so I've not just gone for 10, 9, 8, 7)."
0,0,0,0,25,25,0,25,25,0,"minimum number of castles such that if every castle is won, final score > max score/2 without trying to capture 10, a likely target for high numbers of troops"
0,0,0,0,24,24,0,26,26,0,It's an all or nothing tactic. I chose 4 castles that guarantee victory and concentrated my forces on those. 
0,0,0,0,24,24,0,24,28,0,"I need four wins, and believe 10 and 7 will be more heavily contested"
0,0,0,0,23,24,0,26,27,0,"Half of sum of total points is 27.5, so need 28 to win. Choosing to focus on minimum number of castles (4) needed to get 28 points. Avoiding castle 10 as too popular/risky. Divide troops evenly among castles to contest, but shift one or two soldiers to more valuable castles to gain advantage over same strategy with even distribution of soldiers."
0,0,0,0,22,26,0,26,26,0,"My goal was to get to exactly the minimum 28 victory points while winning the minimum number of castles.  There is no way to do this with three castles, so I must win at least four.  This can be done by winning a variety of combinations, including 1,8,9,10, 2,7,9,10, 3,7,8,10, 4,7,8,9 5,6,8,9, etc.  I chose 5,6,8,9 because I wanted to avoid competition at higher-numbered castles where possible (in this case, avoiding castles 7 and, importantly, 10) thus avoiding strategies that focused on the biggest castle.  I also cannot win without winning at least one castle higher than 6, so ignoring the low numbers is less risky than it may appear.  I then used the 22,26,26,26 distribution to defeat any strategies that evenly distribute across four of the top five castles."
0,0,0,0,22,26,0,26,26,0,It seems like concentrating forces on a few castles is going to be the most successful against a wide variety of strategies.  I then tweaked a little from 25 each to beat anyone else who put 25-25-25-25.
0,0,0,0,22,24,0,26,28,0,"I decided to ignore ties, since any strategy depending on that relies too much on precisely predicting the opponent's strategy. Given that, winning requires winning a minimum of 4 castles with a minimum value of 28. Thus, the objective was to target only this minimum threshold and not allocate any resources to back-ups, contingencies, or disrupting the opponent's strategy. This means assigning soldiers to four towers, and 0 to 6 others. The 10 tower was ignored because it opens up so many options for winning, which also presumes that it will be contended for often. Thus the scenario of taking towers 9, 8, 6, and 5 was chosen. From there, the soldiers were nearly evenly distributed, with a slight bias toward the higher value towers."
0,0,0,0,22,22,0,27,29,0,"28 is the threshold, and there's a lot of ways to make this sum.   Any that uses 4 castles must include 9 or 10, so I picked 9 + .  Picking the ""less desirable after this for other sum sequences seems to favor 8+6+5.   Any troops on castles not needed for victory are wasted.  I propped up 9/8 a little because I figured they'd be more contended, but all 4 castles are equally required for victory."
0,0,0,0,21,21,0,26,31,1,
0,0,0,0,21,21,0,26,31,1,9 + 8 + 6 + 5 = 28. left 1 in 10 for all the people that left it empty
0,0,0,0,20,50,30,0,0,0,"Obviously the castles with the small points do not make a big difference and it makes sense to focus on the larger point castles but at the same time, most people would put more soldiers on the larger castles. So would choose to go where most others would not and still get enough number of points. Assuming that the middle ones do not get enough attention, putting a large number of soldiers here gives a real good chance."
0,0,0,0,20,23,0,27,30,0,"out of 55 victory points, i only need 28 to win the battle.
so i only allot soldiers to castles 5+6+8+9=28."
0,0,0,0,20,20,20,20,20,0,
0,0,0,0,19,23,23,2,0,33,"Wrote a program to choose a random assortment of soldiers, then 'improve' the assortment by moving a soldier from one castle to another, and checking to see if that beats all previous soldier assortments.  It's somewhat surprising that after 244 different deployments of soldiers, it was impossible to come up with a deployment that beat even 55% of the previous deployments with less than 10 million random deployments.  It does appear that deploying soldiers to a few castles is the right thing to do.  Using 33 to try to get the 10th castle, and trying to get castles 5, 6, and 7 with many other soldiers, should be a good plan."
0,0,0,0,19,22,1,28,29,1,"Consider the populations of strategies that will be submitted. Naive strategies include allocating 10 per castle or point weighting the allocation across all 10. Less naive strategies include targeting exactly enough castles to obtain 28/55 points. Some will go after 10-7 with 25 each or point weighted, others will target 7-1, likely point weighted. I put enough on 5-6 to win against the 7-1 point weighted strategy and enough on 9-8 to win against people targeting the large numbers. Of course, this strategy must win against all naive strategies, but probably does not have to beat strategies that would lose to a naive strategy. "
0,0,0,0,18,22,0,28,32,0,Concentrated on winning 28 points (more than 1/2 of the available 55) with the fewest number of castles.  Likely popular strategies will be equal mix or some thing with a little on each castle (more on higher values); this should beat most such strategies.  It would take a big bet on the few castles I made big bets on to beat this.
0,0,0,0,18,21,0,29,32,0,"Minimum number of castles and points required and least popular numbers, weighing the number of troops to the value of the chosen castles."
0,0,0,0,18,21,0,29,32,0,"I picked the smallest number of castles to get 28 points, avoided 10 since I figure a proportion will garrison that heavily, and otherwise picked numbers as low as possible to hopefully face less opposition. I fortified the ones I picked roughly in proportion to their value, as a guess at how well each will be protected."
0,0,0,0,18,0,19,9,24,30,"I made 500 ""random"" strategies and picked the winner from each. Then repeated that 500 times, resulting in 500 ""first-round"" winners, and then submitted the winner from among those.

The 500 first-round winners hopefully somewhat approximate real strategies that real people might pick, and then the winner from among those might approximate an actual winning strategy for the overall game.

Three comments on my solution: (1) the strategy has its simplicity to recommend it, but doesn't think through the psychology much at all; (2) I'm curious but have no idea if this process converges; (3) the two parameters -- 2 rounds, each with 500 entires -- are totally arbitrary, and I wonder how they affect the outcome."
0,0,0,0,17,21,0,29,33,0,"Focusing all on winning 5+6+8+9=28, which is more than half of 1+2+3+4+5+6+7+8+9+10=55"
0,0,0,0,17,21,0,29,33,0,"This is a go for broke strategy attempting to secure a 28-27 victory by taking only 4 castles. Each contested castle receives a number of armies proportional to its value, with the extra 2 units sent to the highest value castles rather than based on simple rounding. "
0,0,0,0,17,21,0,26,36,0,"Same as before, different case."
0,0,0,0,17,19,20,21,22,1,Ten's not worth blowing all your troops for.  Hopefully they wasted too much on ten to match us for 5-9
0,0,0,0,17,17,17,17,16,16,"I assume most people will send more troops to the first castles, and I only need 6 castles to win so If I deploy as many troops as possible to the last 6 and forfeit the first 4 castles I will hopefully dominate the last 6 while opponents waste soldiers on castles I didn't attempt to win in the first place."
0,0,0,0,17,17,17,0,0,49,Optimizer.
0,0,0,0,16,21,1,29,32,1,"Focus on minimum number of castles four.  Focus on the least valued of those. 9,8,6,5 proportionally.   Then adding a  little bit back for possible 10,7.  Little tricky because not working against random troop assignments but working against visitors of 538."
0,0,0,0,16,21,1,28,32,2,"4 castles are highly contested castles picked to add up to 28 points. Troop deployment numbers for these castles are near the 3.57 troop/point ratio that is the maximum number of troops that be deployed to win a point and still win the battle. Chose not to contest lowest 4 point castles at all and put very small amounts in castle 7 and 10 to potentially ""steal"" the points if uncontested or only contested with 1 troop. "
0,0,0,0,16,21,0,31,32,0,"I assume there will be 2 common strategies. Strategy A is to send to each castle soldiers proportional to the amount of points available at the castle, if not a bit skewed toward the higher castles. Something like 24-19-16-13-10-7-5-3-2-1. Strategy B is going all in on just 28 points worth of castles. Something like 40 on Castle 10, 28 on Castle 8, 19 on Castle 6, and 13 on Castle 4. There is little room for error with Strategy B, as losing just 1 of your targets guarantees a loss, but the big advantage here is that all soldiers are warring at the required castles and none are wasted.
I need to figure out a way to beat both of these strategies consistently ---- if I can, I figure I will win enough wars against these two to ignore any other strategies (strategies geared to beat these 2, strategies geared to beat mine, or other ""optimal if both are playing completely logically"" strategies I cannot come up with.) My first idea is to concede Castle 10, giving me more soldiers to play with in the rest of the 9 castles and hopefully proving to be a key advantage going forward. If I was to then proportion my soldiers out similar to Strategy A, but just on the back 9, I would clean up house against Strategy A. However, this will usually doom me against Strategy B. There are so many alterations of Strategy B: 10+9+8+1, 10+9+7+2, 10+9+6+3...39 different ones by my count. Moreover, each castle shows up in 19 to 20 of these different strategies. So I am going to make an assumption that most people who choose Strategy B will choose a 4-castle strategy as that contains the least room for error. The 4-castle strategies are as follows:

10+9+8+1
10+9+7+2
10+9+6+3
10+9+5+4
10+8+7+3
10+8+6+4
10+7+6+5
9+8+7+4
9+8+6+5

10 shows up 7 times, 9 shows up 6, 8 shows up 5, 7 - 4, 6 - 4, 5 - 3, 4 - 3, 3 - 2, 2 - 1, 1 - 1. Still wanting to avoid the assumed-to-be-hotly-contested 10 castle, and noting that all but one contain either a 9 or an 8, I am going to choose my own 4-castle, 28-point strategy that is front-loaded on 9 and 8: 9 + 8 + 6 + 5. Lastly, I chose my distribution of soldiers using a Price is Right-esque strategy, going just over multiples of 5 to hopefully avoid tying with the most common answers. Using complete guesswork, I assume most of these 4-castle strategies will use something like a 40-30-20-10 split, so I put 9 and 8 just over 30 and let 6 and 5 have the rest:
9 - 32
8 - 31
6 - 21
5 - 16"
0,0,0,0,16,17,0,29,38,0,"Picked the smallest group of castles which if all are won gives victory (four castles). Split them to allow for victories in smaller castles (5 and 6) and give up 10 point castle as a hopeful over extension on the enemies behalf. Then split troups, favoring higher point castles."
0,0,0,0,16,16,17,17,17,17,"I played around in Excel with an evolutionary alogrithm, although I'm still not sure it's optimized. "
0,0,0,0,15,20,0,30,35,0,
0,0,0,0,15,19,0,30,36,0,"The easiest way to get to 28 points (the lowest winning score) is to deploy at castles 1, 8, 9, 10. I assume that most puzzlers will figure this out, and so designed a strategy that effectively wins against the ""obvious"" strategy."
0,0,0,0,15,18,0,30,37,0,"Chose 28 points worth of castle, tried to win those."
0,0,0,0,15,17,0,34,34,0,"Trying to win minimum number of castles to win 28 points.  if people go for 10,9,8, they would put roughly 1/3 into each, saving a point to win castle 1.  "
0,0,0,0,15,16,0,33,34,2,"I'm trying to pick my battles, and 9+8+6+5 wins the war.  I hope this beats most people who focus on the top three castles, it makes someone pay for abandoning castle 10, and it should beat people who distribute evenly among the bottom 7 castles."
0,0,0,0,14,20,25,2,2,37,"Main strategy is to go big on winning castles 5, 6, 7 and 10. But I've got a side bet on a few people leaving 8 and 9 either totally undefended or with only 1 soldier."
0,0,0,0,14,16,20,23,27,0,Focused strategy: fighting only for the lowest number of castles required for victory.  Not playing for the most sought-after castle allows more equitable distribution.
0,0,0,0,12,14,16,21,35,2,"This layout beats a lot of typical strategies I could image:
i) 10 on every city
ii) 11-13 on most cities
iii) linear weight on each city (proportional to city strength)
iv) 34-33-33 on any 3 castles
v) 40-30-30 or 35-35-30 on castles 10, 9, 8

We want to either just win a castle, or lose by a lot (so the opponent ""wasted"" a lot of troops). I'm guessing most people will weight castle 10 the most so deploying 2 there feels like it will either just win, or lose by a lot most of the time."
0,0,0,0,11,29,30,30,0,0,"Forfeited 1,2,3,4,9,10 so my max score is 26. decided to distribute evenly across 6,7,8 and throw 10 on 5. Changed from 10 on 5 to 11 in case the highest value is 10's across the board. This is a dumb strategy, but hey we'll see."
0,0,0,0,11,21,0,34,34,0,"All you need are 4 castles totalling 28 (out of 55).  My thought would be that most that recognize this would take the 8-9-10 and any 1 other (most going for either 7 or 1).  But 5+6+9+8 also = 28.  Sending 34 troops each to #8 & #9 would beat those who'd use a 33-33-33-1 deployment with that 8-9-10-x strategy; Sending 11 troops to #5 beats those going for a 10-straight strategy as well as what I call the ""Prevent"" Strategy [of sending (2n-1) troops to each castle with n being the value of each castle].  21 Troops to #6 helps against the 20-even strategy for 6-10, but admittedly fails against a 25-even 4-6-8-10 strategy."
0,0,0,0,11,13,15,18,20,23,"60% of the castles have 82% of the victory points. I just allocated from there. Each one has at least 10, and the specific numbers were pretty random. "
0,0,0,0,11,11,21,26,31,0,
0,0,0,0,10,20,20,20,30,0,"Conceded the low point castles because tier low point value doesn't make them worth defending when 28 needs to be a winning score.  Conceded Castle 10 and instead fortified Castle 9, figuring that most players would go for the top point castle.  Then effectively evenly weighted the remainder of the castles in the hopes that I'd win 3 of the remaining 4 ... and enough points to clear a ""no tie"" threshold."
0,0,0,0,10,14,16,18,20,22,"I figured many people would do the ""optimal"" 13 5 7 9 11 13 15 17 19, and picked a top heavy strategy to beat it."
0,0,0,0,8,16,18,18,20,20,Trial and error led me to think that protecting higher value castles outweighs collecting multiple smaller numbers
0,0,0,0,8,15,20,25,30,2,Because it beat my son. ??_
0,0,0,0,7,21,0,31,41,0,"I am trying to allocate such that I get 28 points in a manner unlikely to be matched by others (passing on 10). Any splits are bonus. I worry that I may have over-allocated to 8 and 9, but am trying to outdo, by 1, any 40-30-20-10 split. "
0,0,0,0,5,12,17,25,23,18,"Didn't waste any on the lower scoring castles. Didn't decide to put a ton on castle 10 either, since it's worth the most and likely will have a lot of soldiers from everybody else. I put almost all of my soldiers at castles 7-10. Almost half of my soldiers are dedicated to 8 and 9, hoping to win either of those plus maybe either 6 or 7 too. Trying to spread out among the higher scoring castles without focusing too hard on one of them, but also prioritizing 8 & 9."
0,0,0,0,5,6,6,11,31,41,To win
0,0,0,0,4,11,21,31,31,2,I have the all the best strategies. 
0,0,0,0,3,5,16,22,27,27,"I wrote some Python scripts to generate random deployments, and compared them against sorted versions, as well as against balanced versions, and I played several rounds where the best ones went on ... not sure if this makes sense as a strategy but it was fun"
0,0,0,0,1,17,19,20,21,22,Emphasized larger castles.
0,0,0,0,1,10,16,20,24,29,Ran a lot of simulated tournaments and picked a top choice from those.
0,0,0,0,1,2,3,2,0,2,"I ran an evolutionary algorithm for a couple hours, and this beat more strategies than any other."
0,0,0,0,0,100,0,0,0,0,"Any selection higher than 6, if a tie, will result in, at most, 5 pts, therefore 6 will win.  "
0,0,0,0,0,33,33,34,0,0,Spooky magic
0,0,0,0,0,29,0,34,37,0,"55 points total. 23 points to win. In order to get 23 points, you need to win at least 3 castles. choose 3 and go ham to win them."
0,0,0,0,0,26,0,35,39,0,"Winning by higher margins is useless in this game.  The only thing that matters is getting 23 points.  I figured that I could either spread the risk and provide multiple ways to get 23 points or put all my eggs in one basket.  With these ""multiple simulation"" games, using a risk spreading strategy basically guarantees you will be in the middle of the pack.  I went with a high risk / greater variability strategy to either win outright, or crash spectacularly. "
0,0,0,0,0,25,25,25,25,0,"Winning 6,7,8,9 will win the battle regardless of the results at the other castles"
0,0,0,0,0,25,25,25,25,0,Guessing -- hoping the odds are in my favor
0,0,0,0,0,25,25,25,25,0,"If I take Castles 6, 7, 8 and 9, I defeat someone who gets the other 6.  It would work, too, with 7 to 10, but I think most people would send many soldiers to 10, and so, I am going to render those moot and enhance my chances of taking 6 through 9."
0,0,0,0,0,25,25,25,25,0,It's a wild guess at a strategy
0,0,0,0,0,25,25,25,25,0,
0,0,0,0,0,25,25,25,25,0,Maximizes points I could win 
0,0,0,0,0,25,24,25,25,1,"A majority is won with only castles 9, 8, 7 and 6, so I deployed 25 to each. Winning these four only will guarantee victory. However, if somebody takes the same approach, they'd beat me. So I reassigned one soldier from castle 7 to castle 10."
0,0,0,0,0,24,24,25,27,0,"If I manage to win 9, 8, 7, and 6, then I'd win the game. 10 seems like the sort of place people might expend a lot of resources over, and the other castles can be forfeited for the sake of those key four. This might backfire against an unexpected number of troops on any of the four, though."
0,0,0,0,0,24,24,24,24,4,"There are a total of 55 points in play here. Assuming no drawn battles, I need to get 28 points (at least) to win. There are 4.2634215e+12 number of solutions to x1 + x2 + ... + x10 = 100. Before I saw this number, I thought for a minute about brute forcing it by finding all solutions and doing a one-vs-one across them all to find the winner. So, I ended up brute-force searching over all solutions of x1+x2+x3+x4 = 20 to get best candidate (0, 5, 5, 10). I ran again for x1+x2+x3+x4=24 to get best candidate (0, 6, 6, 12) and for x1+x2+x3+x4+x5=20 to get best candidate (0, 2, 4, 6, 8). Extrapolating it for the problem of 10 castles with 100 soldiers, I get (1, 3, 5, 7, 9, 11, 13, 15, 17, 19). This (I think/hope/believe) wins on average against the set of all possible solutions but my concern is that I lose against a top-heavy strategy (10, 0, 0, 0, 0, 0, 0, 20, 30, 40) and a bottom-heavy strategy (8, 10, 12, 14, 16, 18, 22, 0, 0 ,0). I ended up going with a middle-heavy strategy giving up on 25 points for castles 1-5+10."
0,0,0,0,0,21,25,25,29,0,You only need 4 castles (excluding 10) to get more points
0,0,0,0,0,21,23,27,29,0,"This is a moneyball strategy.  If I can consistently win castles 6,7,8, and 9, I win every war.  I am placing my bets that opponents will not fortify these 4 castles enough, and am giving up the rest of them in the process.   While I realize that if I lose or tie any of these 4 battles, I lose the war, I think it is a chance."
0,0,0,0,0,21,22,28,29,0,"I only need to conquer castles 6,7,8, and 9 to win a battle."
0,0,0,0,0,20,25,25,30,0,"I previously submitted a solution that was created by a genetic algorithm. Though it was good, I think this is better. (And I created it myself.)

There are 55 points possible, so one needs to score 28 points to win. Since the biggest score with just 3 castles is 10+9+8=27<28, any solution needs to dominate four castles. 

I'm guessing that most people will try to conquer 10 -- so I don't want to dominate that castle. This is just spreading my troops among the next four castles in a fashion that I hope works well.

Thanks for an amazing competition. I'm looking forward to seeing how well my two solutions fare!"
0,0,0,0,0,20,23,27,30,0,"I ran a bunch of simulations to get a feel for winning strategies.  It quickly became clear that winning strategies ignore castles of lower value.  It took longer for me to notice, but every once in a while a strategy that ignored Castle 10 also showed up and did well.

Putting these two together I decided to amass my troops at four castles This is the smallest number of castles that can ensure you get more than half the available points.  Doing so at Castles 6-9 hopefully gives me an advantage over people that expend troops on Castle 10.  I the exact distribution because it worked the best in my simulations (though those are biased based on the mechanism I used to create them).  I also note that this distribution gives me the evenest expected points per troop when I win all four castles."
0,0,0,0,0,20,20,20,20,20,"First five = 15 PT's, next five equal 40 PTs so that deploy troops proportionately bcse  average value is 8."
0,0,0,0,0,20,20,20,20,20,
0,0,0,0,0,20,20,20,20,20,"Need 28 points to win. Assuming that the points happened to coincide with the castle number, I can lose at most one of the castles i bet on and still win."
0,0,0,0,0,19,23,27,31,0,"There is no one answer that can beat everyone.  I decided to focus on 4 castles that would get enough points to win, but I chose not to go for castle 10, because I figured lots of people would try to get 10 or 7-10.  I assigned my troops roughly proportional to the number of points in these four castles."
0,0,0,0,0,19,22,27,32,0,Focused on 6-9 pat castles only
0,0,0,0,0,19,21,25,35,0,"I'll give them castle 10 if it means I'll win castles 9,8,7, and 6. Winning those castles should end with a higher point total. "
0,0,0,0,0,18,19,20,21,22,I can lose one castle and still win a battle
0,0,0,0,0,18,19,20,21,22,I can lose one castle and still win a battle
0,0,0,0,0,16,20,30,30,4,"Winning the top three castles alone isn't enough. You need at least four castles to win. So it seems wise to concentrate troops on four castles that together can take the prize. I figure 10 will be in high demand so I concentrate on 6-9, which together combine for 30 points, more than the 28 points needed to win. 

I put together a program with my son that simulates lots of tournaments of randomly selected deployments and outputs the average of the winners. I was surprised to see a pattern emerge, proving there is some merit to my earlier analysis. Winners tended to put lots of troops in 6, 7, 8, and 9, but fewer in 10. 

I then picked a deployment similar to my final answer and ran it head-to-head against a million random selections, and carefully tweaked it, moving one soldier at a time to find the best possible strategies.  I'm not sure there aren't better strategies -- this may be a local maximum -- but it's good enough. Also, the best strategy really depends on how others play the game. This one does pretty well against soldiers deployed randomly (and another algorithm that assigns soldiers randomly with a  preference for the higher valued castles), but it may not be the best against strategies that are actually employed in this game.  "
0,0,0,0,0,15,15,20,22,28,
0,0,0,0,0,14,15,21,21,29,"I figure if i can deploy enough troops to win at least 4/5 of the 10,9,8,7,6 castle combination and i can win so i am going to guess most other people will put troops in every castle (mistake) so i wanted to fortify my higher level castles with more than most other people might be comfortable with and sacrifice the points for castles 1-5 (15 points) to try to obtain 6-10 (40 points) so even if i lose one of them including a higher number like 10 or 9 i would most likely win the others but i am banking on winning 4/5 of them and if not then i lose but it will be a valiant effort"
0,0,0,0,0,11,14,20,25,30,Get the big ones! 
0,0,0,0,0,11,11,26,26,26,Winning the highest 5 together is far more important than the lower five.
0,0,0,0,0,10,25,65,0,0,"I didn't put too many at the top, I instead put them in the middle where people are less likely to go, but I'll still earn a sizable number of points."
0,0,0,0,0,10,20,30,39,1,"You need 28 points to the war. The fewest castles this can be achieved in is 4, but I can go for #6 instead of #10 and still meet the goal. I'm hoping others overvalue castle 10 and I'm able to win 6-9 for 30 points. If they ignore 10 and contest one of the others integral to my plan I have 1 soldier at 10 to avoid splitting those points"
0,0,0,0,0,10,15,20,25,30,the biggest numbers are the most important x
0,0,0,0,0,6,16,21,26,31,"I anticipated a couple strategies (10 all, proportional distribution, 25 on the top 4, 30-25-20-15-10, etc.) and I tried to come up with a simple strategy that would beat them. I anticipated that many players would use multiples of 5, so I used number that had a remainder of 1 when divided by five to get a slight edge over them. "
0,0,0,0,0,1,12,12,34,41,"Base strategy of 37,33,11,11,1 form castles 10 to 6 with remaining 7 soldiers distributed to give 'optimal' result."
0,0,0,0,0,0,100,0,0,0,Win at least one of the higher value castles
0,0,0,0,0,0,25,25,25,25,you need the last four to win
0,0,0,0,0,0,25,25,25,25,defend the top scoring castles
0,0,0,0,0,0,25,25,25,25,"Submission #3. A variation on my first submission. Here, however, I gave an equal number to each of the castles I defended. "
0,0,0,0,0,0,25,25,25,25,The top four castles are more valuable than the descending 6 combined
0,0,0,0,0,0,22,24,26,28,"I gave up on the smallest three - and then tried to optimally place the others. Not much math involved, honestly. "
0,0,0,0,0,0,21,24,26,29,"Against a random opponent, assigning soldiers in proportion to the value of each castle is a winning strategy. Unfortunately it loses to anyone who focuses troops on a smaller number of castles. Since it is impossible to win without at least one of the four strongest castles, I sent a number of troops proportional to value to each of the four strongest castles, totally neglecting the smallest castles. (I got lazy when trying to calculate the Nash Equilibrium and stopped here.)"
0,0,0,0,0,0,20,24,26,30,Started with a points weighted distribution of troops and created successively better strategies by shifting troops from lower ranked castles to higher ranked castles. 
0,0,0,0,0,0,11,20,27,42,"I'm not convinced of my strategy's effectiveness, so I'm going to submit a few answers to this riddler. In this one I abandoned all of the low value castles and concentrated them all in the highest value castles. I'm not convinced this will work..."
0,0,0,0,0,0,10,20,30,40,1st 6 castles are worth an average of 2 2/3 points.  The last 4 are worth an average of 8 1/2 points.  Any 2 of the last 4 gets more than the first 6 combined.
0,0,0,0,0,0,10,20,30,40,"Trying to win the top 4 castles should outscore all be all who use equal deployment over any 4 or more castles. Also will outscore anyone who just goes for top castle. Will lose to anyone who goes for only top 1, 2 or 3, but they will lose to all equal allocations to more than 3 castles"
0,0,0,0,0,0,10,13,26,51,"This is kind of a variation on (n/2)+1, with a few extra troops thrown in to defeat a potentially like-minded enemy."
0,0,0,0,0,0,0,25,25,50,"My last and final submission!  I ran every distribution of 30 troops / 4 castles to find that [0 7 8 15] performed the best against all others.  For 12 troops / 5 castles = [0 0 3 3 6].  For 10T, 6C = [0 0 1 2 3 4].  I'm extrapolating / guessing that 100T,10C looks like this."
0,0,0,0,0,0,0,0,100,0,I figure many will put all 100 in #10 and thus have lots of ties
0,0,0,0,0,0,0,0,25,75,Because you told me to
0,0,0,0,0,0,0,0,0,100,Go big or go home.  
0,0,0,0,0,0,0,0,0,100,YOLO