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Riddler - Solutions to Castles Puzzle: castle-solutions.csv

This directory contains the data behind the submissions for castles puzzle.

Readers were asked to submit a strategy for the following “Colonel Blotto”-style game:

In a distant, war-torn land, there are 10 castles. There are two warlords: you and your archenemy. Each castle has its own strategic value for a would-be conqueror. Specifically, the castles are worth 1, 2, 3, …, 9, and 10 victory points. You and your enemy each have 100 soldiers to distribute, any way you like, to fight at any of the 10 castles. Whoever sends more soldiers to a given castle conquers that castle and wins its victory points. If you each send the same number of troops, you split the points. You don’t know what distribution of forces your enemy has chosen until the battles begin. Whoever wins the most points wins the war.

Submit a plan distributing your 100 soldiers among the 10 castles. Once I receive all your battle plans, I’ll adjudicate all the possible one-on-one matchups. Whoever wins the most wars wins the battle royale and is crowned king or queen of Riddler Nation!

The data includes all valid submissions, with solvers’ identifying information removed. The 11 columns represent the soldiers deployed to each of the 10 castles, plus a column where the reader could describe his or her strategic approach.

Correction

Please see the following commit: https://github.com/fivethirtyeight/data/commit/c3f808fda5b67aa26ea6fa663ddd4d2eb7c6187f

Data license: CC Attribution 4.0 License · Data source: fivethirtyeight/data on GitHub

1,387 rows sorted by Castle 7

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Suggested facets: Castle 1, Castle 2, Castle 3

Link rowid 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?
1 100 0 0 0 0 0 0 0 0 0 because, I am number one!
5 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.
6 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).
7 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.
8 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.
11 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
13 20 0 0 0 0 0 0 25 25 30 it put high power making it easy to win the castles with troops.
27 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.
38 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).
39 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.
81 11 0 0 0 0 0 0 29 29 31  
99 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.
100 10 0 0 0 0 0 0 30 30 30 Load up the soldiers on the minimum castles needed to win
101 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.
102 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.
131 7 0 0 0 0 0 0 31 31 31 28 or bust.
132 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.
133 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
134 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.
135 6 12 9 15 15 0 0 21 21 1 Picked favorite numbers.
156 6 0 0 0 0 0 0 26 27 41  
201 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.
211 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.
212 5 0 0 0 0 0 0 25 30 40 Only need 28 total pts to win the battle
259 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"
260 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 :(
261 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.
262 4 0 0 0 0 0 0 29 32 35  
263 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.
264 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 yäó_ with x+y =20 as a response to the simplest strategy of 10 10 10 äó_ 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).
268 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.
309 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.
344 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.
345 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.
346 3 0 0 0 0 0 0 29 32 36  
347 3 0 0 0 0 0 0 29 32 36 28 wins, proportional to castle value
348 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.
394 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.
600 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.
601 2 0 0 0 0 0 0 31 33 34  
725 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.
944 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!
945 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…
948 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.
951 1 0 0 0 0 0 0 33 33 33  
952 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.
953 0 25 25 25 25 0 0 0 0 0 HUNGER!
955 0 20 0 20 20 0 0 20 20 0 Guesswork
957 0 17 12 12 12 12 0 35 0 0 Try to capture 28 exact points (majority) without going for castles 10 and 9
958 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.
976 0 10 30 60 0 0 0 0 0 0 swag
978 0 10 0 12 0 20 0 28 0 30 Each even castle is worth more than the odd castle before it
982 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.
983 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.
993 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.
1007 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.
1019 0 4 6 10 15 25 0 40 0 0 Choose the most unlikely combination to be opposed to score 28 points.
1088 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.
1093 0 0 15 15 15 15 0 0 0 40 It beat my previous strategy
1094 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.
1096 0 0 15 0 0 15 0 0 35 35 It beat my previous strategy
1118 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
1124 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.
1125 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.
1126 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.
1131 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.
1156 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 :(
1175 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.
1192 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.
1194 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.
1201 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).
1202 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.
1203 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.
1204 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.
1224 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.
1250 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.
1274 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.
1278 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.
1280 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).
1281 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
1282 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.
1283 0 0 0 0 24 24 0 24 28 0 I need four wins, and believe 10 and 7 will be more heavily contested
1284 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.
1285 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.
1286 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.
1287 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.
1288 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.
1289 0 0 0 0 21 21 0 26 31 1  
1290 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
1292 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.
1296 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.
1297 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.
1298 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.
1300 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
1301 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.
1302 0 0 0 0 17 21 0 26 36 0 Same as before, different case.
1308 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 an…
1309 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.
1311 0 0 0 0 15 20 0 30 35 0  
1312 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.

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CREATE TABLE "riddler-castles/castle-solutions" (
"Castle 1" INTEGER,
  "Castle 2" INTEGER,
  "Castle 3" INTEGER,
  "Castle 4" INTEGER,
  "Castle 5" INTEGER,
  "Castle 6" INTEGER,
  "Castle 7" INTEGER,
  "Castle 8" INTEGER,
  "Castle 9" INTEGER,
  "Castle 10" INTEGER,
  "Why did you choose your troop deployment?" TEXT
)
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