The Binomial Elks Club, being predominantly composed of ambitious and competitive personalities, often held game nights in which members enjoyed battling each other for high stakes and bragging rights. On this particular night, a crowd had gathered around Matt Damon and Ben Affleck. These two friends were engaged in a tense game of intellect, played on a small game board. The rules of this game were:
- There is a 3x3 game board that initially starts with a token in each space.
- Players alternate taking turns. A player's turn consists of taking one or more tokens from a single row or column of the board. The tokens taken are removed from the board. The tokens do not need to be contiguous, just from the same row or column.
- The player who takes the last token wins.
- Each player wagers \$1 million. The winner takes all, the loser gets nothing.
- The order of play is decided by a coin flip. The winner of the coin flip decides whether they will play first or second.
Matt Damon won the coin flip and eventually won the game. Did he choose to play first or second, and what was his strategy?
Having just lost $1 million, Ben wanted to go double or nothing in a rematch. But before Matt could accept, Elon Musk challenged both of them to a 3-person variant of the same game. The rules were slightly different:
- The game board and moves are the same as the previous version. The three players take turns in round robin order, i.e. 1, 2, 3, 1, 2, 3, etc.
- The winner is the player who takes the last token. The player whose turn comes after the winner is 2nd place.
- Each player wagers \$1 million. The winner gets \$2 million, the 2nd place player gets \$1 million, and the 3rd place player gets nothing.
- The order of play is decided by drawing straws. The long straw gets to pick whether to play first, second, or third. The middle straw gets to pick between the two remaining positions. The short straw is stuck playing in whatever position is left.
Ben picked the long straw and Matt picked the middle straw. Which positions did Ben and Matt choose to play, what were their strategies, and how much did each win?
Elon Musk was steaming after losing the previous game. He demanded that the three play a double or nothing match. Matt offered to raise the stakes to \$10 million. Elon agreed on the condition that the straw orders should be reversed from the last game, i.e. Elon picks first then Matt picks second. Matt and Ben gave each other a knowing look and agreed.
Even though Elon chose the same position and played the same strategy as the winner of the previous match, he ended up losing. What happened?
Optional bonus question: Suppose the game board were 4x4 instead of 3x3. How would the answers change for the first two games played? Since the 4x4 strategy can get quite complex, it would be enough to say which position to pick and what the first move(s) should be, rather than list a full tree of possibilities.