Martin Gardner and Wythoff ’ s Game

What’s a question to your answer? We will not settle this puzzle here, yet we’ll taste it. But let’s begin at the beginning, namely in 1907, when Willem Abraham Wythoff [7], a Dutch mathematician, invented the game we analyze here, explained vividly by Martin Gardner in [5]. The game is played with two piles of tokens, each pile containing an arbitrary number of tokens. A move consists of either (i) taking any positive number of tokens from a single pile, possibly the entire pile or (ii) taking the same positive number of tokens from both piles. The player who takes the last token wins. If both players have the same positive number of tokens, the next player wins by taking both piles. It’s called an N position, because the Next player wins. But if both piles are empty, the next player loses, and the previous player, the one who reached the empty piles wins. It’s a P -position, because the Previous player wins. It’s easy to see that the positions (0, 1) and (1, 1) are N -positions, since the next player can win in one move. But (1, 2) is a P -position, since all its followers – positions reached in one move from a position – are N -positions. The first few P -positions are listed in Table 1. Note that every N -position has at least one P -follower. From an N -position a player will move to a P -position in order to win. The order of the two numbers in a pair (An, Bn) is unimportant, but throughout we arrange them in the order 0 ≤ An ≤ Bn.