## Thursday, December 19, 2013

### Nine card Puzzle

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.