Today's puzzle is to analyze a two player strategy game. Nine cards labeled 1 through 9 are placed face up on the table.

Player 1 takes a card.

Player 2 takes a card.

Player 1 takes a card.

Player 2 well you know takes a card.

This continues until either player holds 3 cards which sum to 15. By this I mean exactly 3 cards which sum to exactly 15. Holding 3 such cards is the win condition. i.e. when you pick up that 3rd set making card you win. Alternatively the game could go on until all 9 cards are used up and noone has a winnign set in which case it's a draw.

What happens if both players play perfectly? I initially saw this puzzle in Mathematical Digest, an incredibly awesome mathematics magazine for high-schoolers, with a focus on competition math. As I said it's awesome and this seems like a good time to plug it.

For those of you who have seen it before I'll leave you with trying to figure out which generalizations (if any) are "nice". By generalization I mean having the cards as some multi-set of integers (i.e. allow repeats) and when a win is holding k cards summing to n (choose values of n and k). Possibly allowing multiple pairs of (k_i,n_i) to grant the win.

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ReplyDeleteI have an opinion on this puzzle followed by a non-sequitur. There is a cool interpretation of what the game really is and once you see it it's quite neat. It also makes it easy to determine your strategy. The Cubs will go all the way this year.

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