Sprague-Grundy Values of the $\mathcal{R}$-Wythoff Game

Albert Gu

doi:10.37236/4579

Abstract

We examine the Sprague-Grundy values of the game of $\mathcal{R}$-Wythoff, a restriction of Wythoff's game introduced by Ho, where each move is either to remove a positive number of tokens from the larger pile or to remove the same number of tokens from both piles. Ho showed that the $P$-positions of $\mathcal{R}$-Wythoff agree with those of Wythoff's game, and found all positions of Sprague-Grundy value $1$. We describe all the positions of Sprague-Grundy value $2$ and $3$, and also conjecture a general form of the positions of Sprague-Grundy value $g$.

Highlights

Wythoff’s Game is a two-player impartial game played with two piles of tokens
R-Wythoff is a restriction of Wythoff’s game introduced by Ho [3] where each move is either to remove a positive number of tokens from the larger pile or to remove the same number of tokens from both piles
For the game of R-Wythoff, Ho proved the remarkable fact that the positions of Sprague-Grundy value 0 are exactly the same set as those of Wythoff’s game

Summary

Introduction

Wythoff’s Game is a two-player impartial game played with two piles of tokens. Players alternate turns and for each move a player can remove either a positive number of tokens from one pile, or the same positive number of tokens from both piles. The terminal position has Sprague-Grundy value 0. For the game of R-Wythoff, Ho proved the remarkable fact that the positions of Sprague-Grundy value 0 are exactly the same set as those of Wythoff’s game. We analyze the set of positions of a fixed Sprague-Grundy value for the game

Results

Conclusion