Game positions of Multiple Hook Removing Game

TL;DR

This paper characterizes all game positions in the Multiple Hook Removing Game (MHRG), a combinatorial game on Young diagrams, and establishes equivalences of positions and Grundy values under certain conditions.

Contribution

It provides a complete characterization of MHRG positions and proves an equivalence of positions and Grundy values across different starting configurations.

Findings

Characterization of all MHRG game positions

Equivalence of positions in different starting configurations

Grundy value invariance under certain conditions

Abstract

Multiple Hook Removing Game (MHRG for short) is an impartial game played in terms of Young diagrams. In this paper, we give a characterization of the set of all game positions in MHRG. As an application, we prove that for $t \in \mathbb{Z}_{\geq 0}$ and $m, n \in \mathbb{N}$ such that $t \leq m \leq n$, and a Young diagram $Y$ contained in the rectangular Young diagram $Y_{t,n}$ of size $t \times n$, $Y$ is a game position in MHRG with $Y_{m,n}$ the starting position if and only if $Y$ is a game position in MHRG with $Y_{t,n-m+t}$ the starting position, and also that the Grundy value of $Y$ in the former MHRG is equal to that in the latter MHRG.