Relationship between misère NIM and two-player GOISHI HIROI

TL;DR

This work addresses how misère Nim relates to two-player Goishi hiroi by introducing two functions, $G_{-1}$ and $G^*_{-1}$, derived from misère and normal Nim values. It proves that, for the restricted linear Goishi hiroi game with positions $(x,y,z)$, the normal-play P-positions satisfy $y = G_{-1}(x,z) + 1$ and the misère-play P-positions satisfy $y = G^*_{-1}(x,z) + 1$, with inductive proofs on the total token count $x+y+z$. The paper also provides necessary small-value characterizations via sets $A_0,",A_1,",A_2,",A_3$ and $B_0,",B_1,",B_2,",B_3$, and discusses a 2x2 block structure in $G^*_{-1}$. These results contribute a principled bridge between misère nim analyses and a concrete impartial game, suggesting broader applicability to other rulesets through these oracle-like functions.

Abstract

In combinatorial game theory, there are two famous winning conventions, normal play and misère play. Under normal play convention, the winner is the player who moves last and under misère play convention, the loser is the player who moves last. The difference makes these conventions completely different, and usually, games under misère 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 misère play convention. We also analyze two-player GOISHI HIROI under misère play convention.

Figures (3)

• Figure 1: Puzzle and solution in goishi hiroi
• Figure 2: Position $(x, y, z)$
• Figure 3: Sequence of moves: $(5, 3, 4) \rightarrow (5, 1, 4) \rightarrow (2, 1, 4) \rightarrow (2, 1, 2) \rightarrow (2, 0 , 2) \rightarrow (0, 0, 0)$

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Theorem 1: Sprague Spr35, Grundy Gru39
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5