## Archive for March 3, 2013

### Two Simple Math Games

In his article “What Is the Name of This Game?” author John Mahoney discussed the mathematics of the following game:

Cards numbered 1-9 are placed face up on a table. Two players alternate picking up one card at a time. The winner is the first player who has exactly three cards with a sum of 15.

You can play this game with nine cards removed from a deck of cards, or you can play online by going to http://illuminations.nctm.org/deepseaduel. The online version is a one-player game, but it has modifications that use different numbers of cards, different values on the cards, and different required sums.

Can you find the winning strategy for this game? (Hint: The strategy is described in the linked article above.)

Here’s a modification of the game that seems interesting, too.

Use cards with the following numbers: 1, 2, 3, 4, 6, 9, 12, 18, 36. The winner is the first player who has exactly three cards with a product of 216.

The optimal strategy for this game is different than the strategy from the original game. Can you find it?

**Note:** For the original game, there are eight sets of three cards with a sum of 15:

{1, 5, 9}, {1, 6, 8}, {2, 4, 9}, {2, 5, 8},

{2, 6, 7}, {3, 4, 8}, {3, 5, 7}, {4, 5, 6}

For the modification, there are 36 sets of three cards with a product of 216.

Perhaps that fact will help you identify the optimal strategy.