Solution: The Hundred Years' War
Answer: JOHNSON ASSUMES PRESIDENCY

Written by Jonah Ostroff, Mike Sylvia

The top half of the puzzle is a list of dates and date ranges (labeled “Day 1” through “Day 16”), and the bottom half is a list of questions like “Who won on day 17?” The questions don’t make any sense yet, so let’s start by looking up the dates.

A few of them should yield promising results: 2 September 31 BC was the Battle of Actium, 7 September 1812 was the Battle of Borodino, and so on. Guessing that these are all historical battles should make it possible to identify most of the others.

Many of the resulting battles, like Korsakov, Jargeau, Quebec, Xiapi, and Zurich, start with unusual letters, suggesting that it’s the names that are important rather than the actual historical details. The last letters, while less unusual, also seem to consist of a wide variety of letters. Filling the empty circles in each row with the first and last letters yields the following data:

Day First Battle Last
Day 1: A ACTIUM M
Day 2: W WORCESTER R
Day 3: S STALINGRAD D
Day 4: J JARGEAU U
Day 5: B BORODINO O
Day 6: X XIAPI I
Day 7: L LANGSIDE E
Day 8: K KORSAKOV V
Day 9: F FRANKFURT T
Day 10: Z ZURICH H
Day 11: Q QUEBEC C
Day 12: P PLASSEY Y
Day 13: G GALVESTON N
Day 14: W WAGRAM M
Day 15: R RAVENNA A
Day 16: S SHIPU U

This approach is confirmed by the notes: in the first battle, the circles are A and M, so east “wins” 13-1, and the first thirteen battles contain each letter of the alphabet once.

What’s going on after Day 13? The next letters that appear—W, M, R, A, S, and U—each occurred early on in the list, although not quite in that order. The key observation is that letters which appear together on one day seem to later appear on consecutive days on the same side of the list: Day 1 has A against M, while Days 14 and 15 have M and A on the right. Day 2 has W against R, while Days 14 and 15 have W and R on the left. It looks like the same thing might be happening with the Day 3 and 4 pairs, too, although Day 17’s data isn’t provided.

What could explain this? The flavor text mentions cards, and there’s a card game with this exact behavior: War! In a game of War, players simultaneously flip over a card from the tops of their decks. The player with the higher card takes both cards, and puts them on the bottom of their stack. War is normally played with a 52-card deck, but we’re playing with the alphabet instead. Later letters beat earlier ones, which is why A and M went to M’s side, and W and R went to W’s side. In the examples, and for the rest of the game, the winning card goes before the losing card at the bottom of the deck. (Normally, War also has rules for what to do in the case of a tie. Our deck doesn’t have ties, so we don’t have to worry about that.)

From now on, let’s call the players West and East. The first thirteen plays tell us the original deck order:

West:   AWSJBXLKFZQPG
East:   MRDUOIEVTHCYN

This allows us to simulate the rest of the game, but it’ll be easier if we do a little math first.

Playing out some turns (by hand, using spreadsheet formulas, or with a program) will reveal a few patterns: letters A-G are played every 14 turns, letters H-T are played every 13 turns, and letters U-Z are played every 12 turns. In fact, if we label those letters “L”, “M”, and “H” for “low”, “middle”, and “high”, the two decks are very similar:

West:   LHMMLHMMLHMML
East:   MMLHMMLHMMLHM

After one turn, we get:

West:   HMMLHMMLHMML
East:   MLHMMLHMMLHMML

And after another we get:

West:   MMLHMMLHMMLHM
East:   LHMMLHMMLHMML

This is exactly the original situation, but with the players reversed! After two more turns, we’re right back where we started, except the cards are permuted within each group: the Hs form two 3-cycles, the Ms form a 13-cycle, and the Ls a 7-cycle. So this game repeats every 4*3*13*7 = 1092 turns. The flavor text was right: sometimes War never ends!

Fortunately, 1092 is a small enough number that we can list all possible game states. Here’s a spreadsheet.

With this data, let’s attempt the final exam. Each question asks about the state of the game on one or more days, and the answer is a single letter.

  1. Who won on day 17?
    DayWest’s deckEast’s deck
    17DXILEZHQCWMRAJOBVKTFYPNGUS
  2. Who lost on day 18?
    DayWest’s deckEast’s deck
    18XILEZHQCWMRAOBVKTFYPNGUSJD
  3. Who won on day 75?
    DayWest’s deckEast’s deck
    75HQEXMRCZOIAWTBUPNFVSJGYLKD
  4. Who lost on day 130?
    DayWest’s deckEast’s deck
    130ZMREWOICXTHANBYSJFULKGVQPD
  5. Who fought for the eastern army on day 250?
    DayWest’s deckEast’s deck
    250ZJOAWKTBXPNFSGYILDUHQEVMRC
  6. Who fought for the western army on day 499?
    DayWest’s deckEast’s deck
    499OICZTHAWNMBXJFVLKGYQPDURSE
  7. Who fought for the eastern army on day 1001?
    DayWest’s deckEast’s deck
    1001EXRSCZILAWHQBNMFVJOGYKTDUP
  8. Who fought for the western army on day 3333?
    For days > 1092, we can reduce the day with modular arithmetic. Here, 3333 ≡ 57 (mod 1092).
    DayWest’s deckEast’s deck
    57AZLKBWQPFXRSGOIDYTHEUNMCVJ
  9. Who won on day 99,999,999,999?
    DayWest’s deckEast’s deck
    627SJGXLKDZQPEWRCUOIAVTHBYNMF
  10. Who lost on day 10100?
    DayWest’s deckEast’s deck
    172SBXILFZHQGWMRDUJOEVKTCYPNA
  11. Who fought for the smaller army on day 2500?
    On this question and the next one, one army has 12 cards while the other has 14.
    DayWest’s deckEast’s deck
    316JFXLKGZQPDWRSEUOICVTHAYNMB
  12. Who fought for the larger army on day 12000?
    DayWest’s deckEast’s deck
    1080MDWJOEXKTCZPNAYRSBUILFVHQG
  13. Who was the next person to fight against the loser from day 400?
    DayWest’s deckEast’s deck
    400HFXNMGZJODWKTEUQPCVRSAYILB
    413EXPNCZSJAWLKBHQFVMRGYOIDUT
  14. Who was the next person to fight against the winner from day 1200?
    DayWest’s deckEast’s deck
    108JCWLKAXQPBZRSFYOIGUTHDVNME
    120SFWILGXHQDZMREYJOCUKTAVPNB
  15. Who was stationed in the middle of the eastern army on day 1111?
    This question requires us to look at the composition of the deck, not just the card on top.
    DayWest’s deckEast’s deck
    19ILEZHQCWMRAXOBVKTFYPNGUSJD
  16. Who was stationed in the middle of the western army on day 5001?
    DayWest’s deckEast’s deck
    633DZQPEWRSCXILATHBYNMFUJOGVK
  17. Who was the 12th person to lose by exactly 10?
    This one is easiest to count by hand, since it happens fairly early: O beats E on day 161.
  18. Who was the 4th person to win by exactly 17?
    Similarly, S beats B on day 159.

The last few questions clue words formed by substrings of the decks.

  1. On day 3700, some members of the western army formed a hockey team. Who was its second player?
    DayWest’s deckEast’s deck
    424KBXQPFZRSGWILDUTHEVNMCYJOA
  2. On day 1700, some members of the eastern army formed an organization. Who was its last recruit?
    DayWest’s deckEast’s deck
    608HEZNMCWJOAXKTBVQPFYRSGUILD
  3. Around the 1600th day, some members of the eastern army did something terrible. Who started it?
    DayWest’s deckEast’s deck
    508MBXJOFZKTGWPNDURSEVILCYHQA
  4. Around the 500th day, some members of the western army decided to warm up. Who was stationed immediately behind them?
    DayWest’s deckEast’s deck
    500ICZTHAWNMBXJOFVLKGYQPDURSE
  5. Around the 36500th day, some members of the western army made an announcement. Who was stationed two spaces behind them?
    DayWest’s deckEast’s deck
    464TDZPNEWSJCXLKAVHQBYMRFUOIG
  6. Around the 2000th day, some members of the eastern army created something big. Who was stationed three spaces in front of them?
    DayWest’s deckEast’s deck
    908QGZMRDWOIEXTHCVPNAYSJBULKF

The solutions to the 24 questions spell the answer, JOHNSON ASSUMES PRESIDENCY.

Author’s Notes

This puzzle was inspired by a paper by Mike Spivey.

An earlier version of the puzzle included the question “Who won on day Graham’s number?”, but testers kept skipping it.