The Sprague-Grundy Functions of Saturations of Misere Nim

TL;DR

This paper derives explicit formulas for the Sprague-Grundy functions of saturations of misère Nim, a variant of Nim obtained by adding moves, addressing the lack of such formulas for the original misère Nim.

Contribution

It provides the first explicit formulas for the Sprague-Grundy functions of saturations of misère Nim, expanding understanding of position restrictions in combinatorial game theory.

Findings

Explicit formulas for saturations of misère Nim's Sprague-Grundy functions

Identification of differences between Nim, Welter's game, and misère Nim

Saturation process involves adjoining specific moves to misère Nim

Abstract

We consider mis\`{e}re Nim as a normal-play game obtained from Nim by removing the terminal position. While explicit formulas are known for the Sprague-Grundy functions of Nim and Welter's game, no explicit formula is known for that of mis\`{e}re Nim. All three of these games can be considered as position restrictions of Nim. What are the differences between them? We point out that Nim and Welter's game are saturated, but mis\`{e}re Nim is not. Moreover, we present explicit formulas for the Sprague-Grundy functions of saturations of mis\`{e}re Nim, which are obtained from mis\`{e}re Nim by adjoining some moves.