Wythoff's Game with a Pass

Abstract

This paper describes Wythoff's game with a pass, which is a variant of the classical Wythoff's game. The classical form is played with two piles of stones, from which two players take turns to remove stones from one or both piles. When removing stones from both piles, an equal number must be removed from each. The player who removes the last stone or stones is the winner. In Wythoff's game with a pass, we modify the standard rules to allow for a one-time pass, i.e., a pass move that may be used at most once in a game but not from a terminal position. Once either player has used the pass, it is no longer available. We denote the position of the game by (x,y,p) , where x,y are numbers of stones in two piles and p=1 if a pass is available, and p=0 if not. The authors proved that for (x,y,1) with x <= 9 or y x <= 9 , (x,y,1) is a P-position (the previous player's winning position) if and only if the Grundy number of (x,y,0) is 1 . Therefore, by using the result by U. Blass and A.S. Fraenkel, the Euclid distance between each previous player's winning position in Wythoff's game with a pass and a nearby previous player's winning position in Wythoff's game without a pass is within square root of 20 .