Invariance of the Sprague-Grundy Function for Variants of Wythoff's Game

TL;DR

This paper proves three conjectures about the invariance of the Sprague-Grundy function in certain restricted variants of Wythoff's game, extending previous conjectures and providing new insights into the game's mathematical structure.

Contribution

The paper confirms three conjectures and extends the invariance of the Sprague-Grundy function in specific Wythoff's game variants, advancing understanding of combinatorial game theory.

Findings

Proved three conjectures of Fraenkel and Ho.

Established invariance of the Sprague-Grundy function beyond previous conjectures.

Identified conditions under which the function remains invariant.

Abstract

We prove three conjectures of Fraenkel and Ho regarding two classes of variants of Wythoff's game. The two classes of variants of Wythoff's game feature restrictions of the diagonal moves. Each conjecture states that the Sprague-Grundy function is invariant up to a certain nim-value for a subset of that class of variant of Wythoff's game. For one class of variants of Wythoff's game, we prove that the invariance of the Sprague-Grundy function extends beyond what was conjectured by Fraenkel and Ho.