Equality Classes of Nim Positions under Mis\`ere Play

TL;DR

This paper classifies Nim positions into equivalence classes under misère play, revealing that in impartial contexts only trivial identifications occur, while in partizan contexts all Nim positions remain distinct.

Contribution

It provides a complete characterization of misère equivalence classes of Nim positions under two different equivalence relations, extending understanding of misère combinatorial game theory.

Findings

Impartial context: only known equivalences from adding a heap of size 1.

Partizan context: all Nim positions are inequivalent.

Clarifies the structure of Nim positions under misère play.

Abstract

We determine the mis\`{e}re equivalence classes of Nim positions under two equivalence relations: one based on playing disjunctive sums with other impartial games, and one allowing sums with partizan games. In the impartial context, the only identifications we can make are those stemming from the known fact about adding a heap of size 1. In the partizan context, distinct Nim positions are inequivalent.