Mathematics Problem Of the Week

Spring 2002

POW #3

Countdown

Countdown is a two-player game played on a 2-dimensional grid consisting of all positive integer points. A marker is placed at a random point (m,n) on the grid. At each turn a player may move any number of spaces to the left (i.e. decrease m by an integral amount) or any number of spaces down (decreasing n), but may not move off the board. Thus the new point (m*,n*) must have either 0<m*<m and n*=n or m*=m and 0<n*<n. To win the game, you must leave the marker in a position where your opponent has no legal move.

(a) If you can choose whether to move first or second, can you find a strategy that will always allow you to win? If so, explain the strategy. If not, explain why such a strategy is impossible.

(b) For the more spatially inclined - what happens if you move the game to three dimensions (with the ability to count down in any one dimension during each turn)?

BONUS: For the hyper-spatially inclined, how would you approach the game of Countdown in n-dimensions?

Due Friday, February 15^th at Noon.