Variants of Wythoff's game translating its P-positions

TL;DR

This paper introduces F-Wythoff, a variant of Wythoff's game, characterizes its P-positions as shifted original P-positions, and explores the distribution of Sprague-Grundy values, including differences in normal and misere versions.

Contribution

It defines F-Wythoff, shows P-positions are shifted Wythoff P-positions, and analyzes Sprague-Grundy distributions and differences between game variants.

Findings

P-positions of F-Wythoff are shifted Wythoff P-positions by adding 1.

Distribution of Sprague-Grundy values is characterized and generalized.

Normal and misere versions differ only on positions with Sprague-Grundy values 0 and 1.

Abstract

We introduce a restriction of Wythoff's game, which we call F-Wythoff, in which the integer ratio of entries must not change if an equal number of tokens are removed from both piles. We show that P-positions of F-Wythoff are exactly those positions obtained from P-positions of Wythoff's game by adding 1 to each entry. We describe the distribution of Sprague-Grundy values and, in particular, generalize two properties on the distribution of those positions which have Sprague-Grundy value k, for a given k, for variants of Wythoff's game. We analyze the misere F-Wythoff and show that the normal and misere versions differ exactly on those positions which have Sprague-Grundy values 0, and 1 via a swap. We examine two further variants of F-Wythoff, one restriction and one extension, preserving its P-positions. We raise two general questions based on the translation phenomenon of the P-positions.