Problem
of the
Week
PROBLEM 133

Note:  Problems this semester will be posted by 3:00pm on Mondays.

Here's a game for two players to play.  You and your opponent are given a point (n,m) chosen at random from the first quadrant of the coordinate plane.  The two of you then alternate turns; when it's your turn, you may make one of either type of the following moves:

(L)  Replace the present point by a new point at least one unit away and to the left, and in the first quadrant, i.e. replace (n,m) by (n',m) with 0 < n' £ n - 1;

or

(D) Replace the present point by a new point at least one unit away and below, and in the first quadrant, i.e. replace (n,m) by (n,m') with 0 < m' £ m - 1.

The first player who cannot make a legal move loses.

Do you want to go first or second, and what is your winning strategy?

For the more spacially inclined: What happens if you play the game in three-dimensional space allowing moves similar to the ones above, but in any one of the three coordinate directions?

For the hyper-spacially inclined: Play the game in n-dimensional space.

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ã2002 Alberto L. Delgado