Wythoff's Game with a Pass

TL;DR

This paper studies a variant of Wythoff's game allowing a single pass move, analyzing how the pass affects winning positions and relating it to the classical game's structure.

Contribution

It introduces and analyzes Wythoff's game with a pass, establishing conditions for P-positions and relating them to the classical game's Grundy numbers.

Findings

P-positions with a pass are characterized by the Grundy number of the standard game.

The pass significantly alters the structure of winning positions.

Euclidean distance bounds between P-positions with and without a pass are established.

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 .