Riddler - Solutions to Castles Puzzle: castle-solutions-3.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

1,466 rows sorted by Castle 10

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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?
60 3 6 9 14 18 22.0 28.0 0.0 0.0 0.0 Ignore the top ones, focus on minimum needed for majority of points
91 17 17 17 17 17 5.0 0.0 0.0 0.0 0.0 overcome 6
105 1 5 0 7 8 21.0 0.0 28.0 30.0 0.0 optimize higher castles but never go in increments of five (leads to more ties which are inefficient). use 0 on castles that have a higher chance of being contested
110 1 4 6 14 18 24.0 33.0 0.0 0.0 0.0 Assuming more valuable castles will be more contested, negating their points advantage. 28 points wins, so it only makes sense to contest castles worth that many total. I took 1-7 (28 total pts), with troop allocations focused on the hotly contested 5,6,7 castles. I'm hoping to 'pay' for those by taking 1,2,3 cheaply.
116 1 0 9 0 0 10.0 10.0 20.0 40.0 0.0 Adjustments to previous contest
131 1 0 0 0 0 0.0 0.0 0.0 99.0 0.0 Just Cause
141 1 0 0 0 0 0.0 0.0 99.0 0.0 0.0 You need to get points, and probably the only way to do that is to win a house outright. I am guessing that someone will do 100 for 10 and 9, so guessing 8 will be the one where people don't apply 100.
147 1 0 9 15 0 20.0 25.0 30.0 0.0 0.0  
148 1 0 0 4 11 14.0 21.0 26.0 24.0 0.0 I started with zero at Castle 10, and a large chunk (25) at 8 and 9. I then gave 5 fewer troops to each Castle going down until I ran out. Then I went back and added in a bit of noise. Then I noticed it required >0 for Castle 1, so I put that in.
191 2 3 4 0 6 15.0 10.0 26.0 34.0 0.0 Clustered to win as many points against last time's winners.
199 1 0 0 2 21 22.0 3.0 24.0 27.0 0.0 Key is to get to 28. Wanted to stack as few castles as possible to increase probability of winning those. Left 7, 4, and 3 as contingency plans in case someone was doing the same.
205 21 18 15 12 12 10.0 9.0 6.0 0.0 0.0 I made three simplifying assumptions about my opponents' strategies: first, they want to hold as few castles as possible to get over the victory threshold; second, they understand that it is a waste to have more than the threshold of victory points needed; third, they ascribe the same strategic value to each of those castles, as their strategy fails without any one of them. This means that my average opponent will aim to hold four castles, worth 28 victory points and will deploy 25 troops to each. There are (by my very quick, admittedly) count, 9 unique strategic combinations of four castles that get to the victory threshold. I assume that my opponents are indifferent about which one they choose and arrive at whichever one they wish to play randomly. I use the frequency with which a castle worth a given number of victory points appears in one of the 9 unique four-castle strategies to generate the probability that my average opponent, within my simplifying assumptions, would place troops at that castle, and subsequently, how many soldiers (on average) I expect to be stationed at that castle. I would then simply distribute my 100 soldiers so I had marginally more at each castle than my opponent. Noting the inherent risk of this strategy (every battle should be a draw if my opponents play as I do, or as I expect them to give or take a trembling hand or two), I (rather randomly) decide that the castles worth 1 or 2 victory points are of low strategic value, given how infrequently they are included in 4-castle strategy and redistribute the six troops I would have placed there in the purer form of my strategy to the castles worth 10, 9, 8, 7, 6 and 5 VPs. Hooah!
225 1 1 0 25 0 0.0 25.0 25.0 25.0 0.0  
232 1 5 5 5 20 20.0 20.0 20.0 0.0 0.0  
238 1 1 0 9 14 20.0 25.0 30.0 0.0 0.0 Just give up on the biggest ones, probably a waste
249 1 0 0 12 0 12.0 25.0 25.0 25.0 0.0 In order to assign the maximum number of soldiers to selected castles, from all castle combinations that sum up to 28 with just 4 castles, I choose to ignore castle 10 and concentrate forces to 9,8,7 (25 on each) then I just need one of 4,5 or 6 so I had to share the rest 25 soldiers to those 3 castles. To increase chances I placed 12 soldiers to 4 and 6 and the last remaining to castle 1( that was unintentional, since I had to place at least on soldier to castle 1)
266 2 4 6 7 8 15.0 23.0 35.0 0.0 0.0 Idk, could work
290 4 7 5 21 21 12.0 20.0 7.0 3.0 0.0 Took average of top 5 winners from first battle, average of top 5 winners from second, and guessed the trend of the top 5 from this battle would look like [0, 0, 0, 15, 16, 0, 0, 0, 39, 30]. Used evolutionary machine learning to find a strategy that would consistently give highest scores against slight variations on the predicted opponent strategy.
302 7 8 9 10 11 12.0 13.0 14.0 16.0 0.0 Sac the queen!
315 1 0 0 9 0 15.0 0.0 35.0 40.0 0.0 Cheapest way to 28 total points. It did make me place one troop in castle one for some reason. Would rather have put that soldier at 4.
317 7 9 9 11 13 0.0 0.0 0.0 51.0 0.0 It adds up to >20 points and I don't think anyone's gonna care as much as I do about the ones I chose? Idk though
330 1 2 5 5 7 20.0 20.0 20.0 20.0 0.0 Sacrificed Castle 10 in hopes of winning slightly-lesser castles
341 3 0 7 10 20 0.0 30.0 30.0 0.0 0.0 I targeted 6 castles that would get me 28 points. If I go 6/6 on those ones that I bet big on then I win (doesn’t really feel like a good strategy, but I wanted to see how it would play out)
364 5 7 8 10 15 25.0 30.0 0.0 0.0 0.0 Willing to concede three castles with most points in hopes of winning all others (28 of 55 possible points). Assigning most soldiers to those with most points among the group that I was aiming to win.
366 0 0 0 14 17 20.0 23.0 26.0 0.0 0.0 Ignored 9&10 and chose the fewest castles past that to give me more than 28 points and weighed troops by value
369 0 0 0 13 15 18.0 26.0 28.0 0.0 0.0 Distributed my troops evenly through 4-8 which will give me 30 points each time banking on that I have more troop in those stations giving the other opponent 10-9-3-2-.
375 12 12 12 12 12 14.0 26.0 0.0 0.0 0.0 I figure the bulk will put their points in to the top 4 if i can win everything else i should be good to go
385 0 0 0 0 20 23.0 0.0 30.0 27.0 0.0 There's no way to win without at least four castles, so I focused on winning four and tried to optimize versus earlier distributions.
387 4 5 6 12 21 26.0 26.0 0.0 0.0 0.0 I did the math and discovered that 28 points is the magic number. 8, 9, 10 get you 27, and 1-7 get you 28. So, I punted on 8,9,10, expecting most people to stock up on those and give them a free victory there while they use the majority of their troops. Meanwhile, I'll be happy to take all the smaller castles because 28>27. I debated going for 8,9,10 and 1 to take 28 points, or even 2,3,4,6,7,8 to make 28, but figured my first thought would win more often than the other two, which would be harder to distribute troops since 8 would take so many to guarantee the victory.
394 0 1 1 10 0 0.0 29.0 29.0 30.0 0.0 Exact victory points, fewest required wins, avoid 10.
395 0 0 1 2 20 22.0 3.0 24.0 28.0 0.0 Resubmission of my last entry, which required me to put at least one on castle 1. Want to concentrate my efforts on reaching 28, the required score for winning the battle. The others are slight contingencies, in case someone else does the same thing.
405 0 0 11 12 17 0.0 25.0 0.0 35.0 0.0 I need 28 points to win, castle 1 and 2 have little value, I feel like people will value 10 and or 8 highly. 10 seems like a median number and something someone would throw at 3 or 4 so I went with 11 and 12. It's really a win all or lose scenario for me. Hopefully people spend resources out instead of concentrating. 10,9,8,1 seems like the most common strategy for people to really go after, I think I can overwhelm the 9 slot and forfeit the others while getting what I want
406 0 11 11 12 12 13.0 13.0 14.0 14.0 0.0 This won't work, but I am attempting to avoid over-optimisation by ignoring all previous data. Accept the loss of 1 and 10, and try to win on average against the rest, with a slight bias to higher value targets
408 0 0 0 0 0 0.0 0.0 0.0 100.0 0.0 Nash Equilibrium
442 6 6 5 15 20 20.0 28.0 0.0 0.0 0.0 Seed the top scoring castles and focus heavy on winning the middle ones. The castles worth few pointe I assumed few people would go for
448 8 12 13 13 13 14.0 0.0 27.0 0.0 0.0 I hope to allow my opponent to take the top two and the 7th castle while preserving those forces to have enough to counter what I expect to be a smaller amount dedicated to castles 1-6 and 8, thereby getting a majority of points and castles.
460 0 1 1 17 20 20.0 20.0 20.0 1.0 0.0 Dominant the middle/paint like in basketball
500 0 0 0 11 0 0.0 26.0 31.0 32.0 0.0 I went for the less "psychologically significant" castles which would still give me a significant advantage. I sent 11 troops to 4 as an additional bonus in case someone is close to me in the upper ranges, or sweeps all the castles I didn't send any troops to - and since 11 just barely beats the simple strategy of sending 10 troops to each castle. I sent 26 to 7 because 26 is one more than 25 (another round number I expect people to use a lot), and similarly I sent 31 (rather than 30) to #8. Hope this works!
511 0 0 0 0 20 50.0 30.0 0.0 0.0 0.0 6 seems like a good number. And I didn't want to send any lone soldiers off to die. I expect to win Castle 6 around 1/3 of the time, so hey, that's like 2 points. I'm feeling positive about it.
543 0 0 0 0 0 25.0 0.0 34.0 41.0 0.0 The minimum number of castles needed is 3 which have to add up to 23. 6 is app. 25% of 23 so 25 soldiers 8 is app. 33% of 23 so 34 soldiers and the rest go to 9.
545 4 4 4 7 5 25.0 25.0 25.0 0.0 0.0 I assume most opponents would direct the greatest resources to the biggest castles, possibly also directing more substantial ones towards those in the middle of the bracket (5 and 6). While I will lose 9 and 10, opponent investments there should enable me to hold 6 ,7 and 8, which would give me a 2-point advantage at the top range. By dedicating some resources lower I think I'm more likely to gain and hold 1-4 even if I lose 5. (I think 7 soldiers are more likely to win 4 than 5, and if I take some of the lower castles I don't care anyway.)
558 1 3 4 7 13 20.0 24.0 28.0 0.0 0.0 I figured most people would choose increasing sequences, which means a lower numbers on 1-8 and more on 9 and 10. So if I put all my solders on 1-8 and beat them, maybe I'd have a better chance! :)
569 0 0 10 15 15 15.0 15.0 15.0 15.0 0.0 rather take the sum of the middle numbers over the first and last
573 0 10 0 0 15 25.0 25.0 25.0 0.0 0.0 Only deploy to certain castles to win, hope to get lucky.
578 1 1 4 4 10 20.0 20.0 20.0 20.0 0.0 Avoid wasting resources on a high contention battle (Castle 10). Spread out on high value targets with less contention (Castle 9 through 6).
580 0 0 0 15 15 15.0 25.0 30.0 0.0 0.0 Play for the middle and push for the top but don’t over commit
582 0 0 0 0 16 19.0 0.0 30.0 35.0 0.0 I'm going all-in for getting the bare minimum points of 28 or more. The fewest castles I need is 4. 10-9-8-7 is an option but lots of people will go after castle 10, so I'm going after 5-6-8-9. Same number of castles, but I'm playing off the beaten path. Also, 5-6-8-9 are all castles that are in fewer winning combinations, so they're more likely to be won by me. The actual troop placements are based on the relative difficults I computed for winning those particular castles.
590 4 7 10 14 18 22.0 25.0 0.0 0.0 0.0 Get 28pts by focusing on the less valuable castles
597 1 1 2 5 20 20.0 20.0 20.0 30.0 0.0 Monte Carlo simulation, I think, with troops being weighted toward the higher-point castles with an inexact strategy picking a random number between 0-100 for the 10th castle and randbetween 0-remaining troops in the 9th and so on until the 1 point castle. simulated this 3000 times, then maximized my point gap between the average results with some buffer troops thrown in.
605 0 4 6 8 11 14.0 22.0 23.0 12.0 0.0 I'll never tell.
614 0 0 25 0 25 0.0 25.0 25.0 0.0 0.0 Sacrifices must be made! Castles 1, 2, 4, 6, 9, and 10 are dead to me! Going hyper-aggressive (but not the most aggressive strategy). Best Case: I win! Worst Case: I am a troll!
621 0 4 0 0 22 22.0 22.0 30.0 0.0 0.0  
623 0 0 0 0 17 21.0 0.0 26.0 36.0 0.0 I think a lot of people will be fighting for #10 and #1 because 10 is worth the most points and #1 is the tiebreaker if you went 10,9,8,1 or 7,6,5,4,3,2,1. I considered going for 10,9,8, 2 to avoid fighting over the #1 and because I could win even with a tie on #2, and then realized I could avoid #10 as well. In summary, I'm avoiding fighting over what I expect to be hotly contested #10 and #1 in favor of #6 and #5 while maintaining the concentration of my troops by only needing to capture 4 castles to win. As far as specific troop distribution goes, I made sure I had at least three times the castle number and dumped a bunch extra on #9, which I think will receive a heavy designation from anyone pursuing a variant of the 10,9,8,1 strategy. I did not assign any troop numbers that end in 0 or 5, they are too popular.
648 0 0 0 0 21 21.0 0.0 29.0 29.0 0.0 Let me try this again because I did my math wrong. Sacrifices must be made! Castles 1, 2, 3, 4, 7 and 10 are dead to me.
666 0 2 0 11 11 0.0 25.0 24.0 27.0 0.0 You only have to win by a little.
668 0 0 0 15 15 20.0 25.0 25.0 0.0 0.0 Focus more troops on enough points to get more than half of points.
670 0 4 4 4 8 0.0 24.0 26.0 28.0 0.0 I feel like putting a lot at 10 is risky, because a lot of people will put a lot at 10 and a loss is devastating. I loaded up on castles 7, 8, 9, gave up on castle 6 and 1, and dispersed the rest.
675 0 6 8 10 11 12.0 13.0 14.0 15.0 0.0 I looked at how much more valuable on average each castle is to the others below it and sent troops based on this calculation normalized and rounded for
678 0 4 8 10 15 5.0 30.0 23.0 5.0 0.0 Putting more troops into the medium level castles
679 3 6 0 14 0 22.0 25.0 30.0 0.0 0.0 I figured you need 28 points to win and winning 1-7 will get you there exactly. That means you can reallocate all your points from 8-10 to 1-7 and stand a good chance of winning. Other people might do that too though, so I did some other stuff on a whim to mix it up.
699 4 7 10 14 17 22.0 26.0 0.0 0.0 0.0 Distributed proportionally-ish on the buckets (hopefully) most likely to get to 28
709 0 0 5 15 5 10.0 20.0 20.0 25.0 0.0 I abandoned the first and last castles as not worth fighting over and focused on castles a little before and after the center that other teams might neglect.
716 0 0 0 0 0 100.0 0.0 0.0 0.0 0.0 All of the troops at the first castle higher than 5
719 0 1 17 25 10 9.0 9.0 9.0 6.0 0.0 I looked at old answers and fudged a little honestly.
732 0 0 0 0 0 0.0 26.0 32.0 42.0 0.0 I only need to win 3 castles, assuming people focus on 10, I decided to ignore it an focus on the next three and then power creep 9 and 8 in case people had the same idea as I did.
739 0 0 8 11 0 22.0 28.0 31.0 0.0 0.0 Strongly attacked with the most likely castles to reach 28.
741 0 0 0 0 23 24.0 25.0 0.0 28.0 0.0  
749 0 8 9 10 13 15.0 20.0 0.0 25.0 0.0 When you consider how many soldiers you spend for each point gained, from the previous data eight is the worst value, so should not be contested and ten is the best value, so I think many people will be trying to prioritize castle ten, so I just left it out. Victory doesn't come by contesting all the points but by being able to secure more than half of them. basically 49% of the points don't matter at all.
758 1 2 2 2 2 21.0 22.0 23.0 25.0 0.0  
768 1 1 1 1 1 2.0 31.0 31.0 31.0 0.0 all out to capture 7,8,9 and pick up any 0s elsewhere
774 1 2 5 13 17 0.0 26.0 0.0 36.0 0.0 I need 28 VPs. So I aimed for an unusual combination of getting them. As long as I get castles 3, 4, 5, 7 and 9, I have my 28 points and have no need to get any others. I will lose only to people who outbid me on one of these five, but those who don't bid 0 on any, or even multiple, castles, will have fewer troops to deploy on those five, so my chances are reasonably good. I expect to lose to those who max out on castles 9 and 10 but to win against a good percentage of other contestants. I made a late change to go for 3+ points from 1, 2 and 3 combined
795 4 6 8 12 17 22.0 31.0 0.0 0.0 0.0 Focus on the front 7, which adds up to 28, which gives you one more than your opponent, who takes 7,8,9 (total 27)
798 0 7 0 14 0 21.0 25.0 0.0 33.0 0.0 I considered strategies which are most efficient in usage of troops (ie. trying to get exactly 28 points) which would allow for ~3.57 troops per point value of the castle. Then I considered rounding error on the troops deployed - if others are also using 28-point strategies, then the best of them would be those that used the castles with small negative rounding errors. (ie. Castle 2 asks for ~7.14 troops but would be satisfied with 7). So I pick castle 2,4,6,7,&9 which leaves me with one leftover troop - I think Castle 9 might be the most competitive among 28-point strategies, so I drop the extra troop there.
801 10 10 10 10 10 25.0 25.0 0.0 0.0 0.0 There are 55 points up for grabs. To win, I would need 28 or more. I disregard castles 8, 9, and 10. That loses me 27 points. However, I deploy the remaining soldiers in the following manner - 1. Castles 6 and 7 get 25 soldiers each. Assuming that the opponent has committed most soldiers to castles 8, 9, and 10, I should be able to gain these two castles. 2. For the remaining castles, I will assign 10 soldiers each. The hope is that the opponent over-commits on the higher value castles while undervaluing the remaining castles. By flipping that thinking on its head, I hope to undermine the opponent's strategy.
808 0 0 8 10 12 14.0 17.0 19.0 20.0 0.0 I guessed that an distribution proportionate to point values will rarely win the 10 and will waste trips on the low-value castles, so I dropped the 10 and the bottom too and then loosely distributed them proportionally from there fight estimating as I wrote on some construction paper with a crayon.
817 0 2p 0 0 20 20.0 20.0 20.0 0.0 0.0 Try to get to 28 in a way that average person wouldn't do.
821 11 11 11 13 14 20.0 20.0 0.0 0.0 0.0 I think people will underinvest in low value castles, and invest more on high value castles than the middle range ones. So my hope is to win one through five relatively cheaply, while having a decent chance of winning 6 and 7.
855 2 2 2 8 0 19.0 26.0 41.0 0.0 0.0 Avoid wasted troops at high value targets and low v; win on aggregate over sim.
857 0 0 0 5 9 14.0 21.0 21.0 30.0 0.0 Ill sacrifice the extremes and try to take the bulk of the points in the middle
871 0 0 0 0 15 20.0 0.0 40.0 25.0 0.0 Choose four castles whose total point value is 28. Go all out for them.
887 0 0 0 5 7 10.0 21.0 24.0 33.0 0.0 Avoided overcommit on 10. Attempted to stack 9 and upper middle.
911 0 0 2 30 2 30.0 2.0 34.0 0.0 0.0 Three eyed raven told me
915 0 0 0 0 0 19.0 23.0 27.0 31.0 0.0 All focused on the fewest castles needed to win, avoiding the highest and lowest valued.
928 0 0 0 0 0 17.0 18.0 30.0 35.0 0.0  
929 0 0 7 10 12 14.0 17.0 19.0 21.0 0.0 1 and 2 are low-value; 10 will be too heavily contested
941 1 0 0 0 0 0.0 30.0 30.0 39.0 0.0 Highest % troops outside Castle 10
955 0 0 0 20 0 0.0 26.0 26.0 28.0 0.0 Maximizing distribution to minimum number of castles needed to win, while avoiding expense of castle 10.
964 5 5 5 10 20 25.0 30.0 0.0 0.0 0.0 trying for a plausible counter-intuitive plan
967 2 3 4 5 8 12.0 16.0 24.0 26.0 0.0 hit the higher valued castles harder, except for 10, which I believe my opponent will overvalue.
971 2 4 6 6 6 21.0 25.0 30.0 0.0 0.0  
979 2 2 2 2 12 20.0 25.0 35.0 0.0 0.0 I didn't try for 9 or 10 and went for 5-8.
987 0 10 0 0 0 0.0 15.0 25.0 50.0 0.0 Forces concentrated on alternative 4 castles to win
993 0 0 0 0 19 23.0 0.0 27.0 31.0 0.0 Go all-in on 4 castles that give just enough points to win (28), ceding the other 27 points’ worth. Stack a few more troops on the high value castles just because.
1011 11 11 11 12 14 15.0 16.0 0.0 0.0 0.0 I went for 28 out of 55 points by selecting the lowest values that add to 28.
1012 3 7 10 14 18 22.0 26.0 0.0 0.0 0.0 I aimed to win 28 points (minimum for a simple majority out of 55), and targeted the lowest value castles to reach a 28-point total while avoiding committing troops to the high-value targets. My goal was to pay just over 3 troops per point.
1015 2 5 10 10 15 15.0 20.0 23.0 0.0 0.0 Trumpian Electoral college: ignore NY and CA, go for TX, PA, FL
1019 11 11 11 11 14 21.0 21.0 0.0 0.0 0.0 I expect most people to put most of their troops in the higher numbered castles, so my strategy is to win the lowest 7.
1020 1 1 1 11 11 20.0 25.0 30.0 0.0 0.0 castle 9 and 10 would be the most valuable so should get the largest number of troops assigned to them by the other overlords so fighting over them would be the most pointless allocation of troops since you're most likely to lose there. castles 1-3 are of limited value so while they could safely be ignored you could steal one of them with minimal troop numbers. combining those 5 castles gives you 25 points which won't be enough to win. castles 6-8 are the most valuable as far as being high enough to want to take but not so high that you would risk sending all your troops to, so 20-30% of your forces should be enough to win those three, especially castle 8 as you've conceded 9 and 10 already so you have to win castle 8 . castles 4 and 5 are the risky ones as losing either one means you lose, but again aren't valuable enough for large troop dispositions. however in the event of the enemy dividing his troops evenly among all 10 castles I need to commit more than 10 troops to ensure victory. doing things this way should give me a 30-25 victory
1024 2 4 6 12 16 18.0 20.0 22.0 0.0 0.0  
1025 0 0 0 20 20 20.0 20.0 20.0 0.0 0.0 Why not?

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