Relationship between mis\`ere NIM and two-player GOISHI HIROI

TL;DR

This paper explores the relationship between misère NIM and two-player GOISHI HIROI, showing how to determine the winner under normal play using misère NIM and analyzing GOISHI HIROI under misère rules.

Contribution

It establishes a novel connection between misère NIM and GOISHI HIROI, enabling winner determination across different game conventions.

Findings

Winner of GOISHI HIROI under normal play can be determined via misère NIM.

Analysis of GOISHI HIROI under misère play provides new insights.

The relationship simplifies complex game analysis in combinatorial game theory.

Abstract

In combinatorial game theory, there are two famous winning conventions, normal play and mis\`ere play. Under normal play convention, the winner is the player who moves last and under mis\`ere play convention, the loser is the player who moves last. The difference makes these conventions completely different, and usually, games under mis\`ere play convention is much difficult to analyze than games under normal play convention. In this study, we show an interesting relationship between rulesets under different winning conventions; we can determine the winner of two-player GOISHI HIROI under normal play convention by using NIM under mis\`ere play convention. We also analyze two-player GOISHI HIROI under mis\`ere play convention.