Two variants of Wythoff's game preserving its P-positions

TL;DR

This paper introduces two variants of Wythoff's game that preserve its P-positions, providing formulas for positions with Sprague-Grundy value 1 and analyzing the Sprague-Grundy functions.

Contribution

It presents two new variants of Wythoff's game that maintain original P-positions and offers formulas and analysis for their Sprague-Grundy values.

Findings

Both variants preserve Wythoff's P-positions.

Formulas for positions with Sprague-Grundy value 1 are provided.

Several results on the Sprague-Grundy functions are proved.

Abstract

We present two variants of Wythoff's game. The first game is a restriction of Wythoff's game in which removing tokens from the smaller pile is not allowed if the two entries are not equal. The second game is an extension of Wythoff's game obtained by adjoining a move allowing players to remove k tokens from the smaller pile and l tokens from the other pile provided l < k. We show that both games preserve the P-positions of Wythoff's game. This resolves a question raised by Duchene, Fraenkel, Nowakowski and Rigo. We give formulas for those positions which have Sprague-Grundy value 1. We also prove several results on the Sprague-Grundy functions.