In memory of Martin Gardner, who was and remains enchantingly influential and inspiring.

WYTHOFF

In 1907, the Dutch mathematician, Willem Abraham Wythoff [13] invented this game, later vividly explained by Martin Gardner in [7].

Wythoff is played on a pair of nonnegative integers, (M, N). A move consists of either (i) subtracting any positive integer from precisely one of M or N such that the result remains nonnegative, or (ii) subtracting the same positive integer from both M and N such that the results remain nonnegative. The first player unable to move loses.

Given the position (3, 3), say, the next player wins in a single move: (3, 3) → (0, 0). The position (3, 3) is called an N-position, because the Next player wins. If M = N = 0, the next player loses, and the previous player, the one who moved to (0, 0), wins. Thus (0, 0) is a P-position, because the Previous player wins.

If M > 0, it is easy to see that (0, M) and (M, M) are N-positions, since the next player can win in one move. On the other hand, (1, 2) is a P-position because all its followers—positions reached in one move—are N-positions. The first few P-positions are listed in Table 29.1. Note that every N-position has at least one P-follower, but all followers of a P-position are N-positions. From an N-position, in order to win, a player must move to a P-position.