Impartial Triangular Chocolate Bar Games

TL;DR

This paper explores the mathematical structure of triangular chocolate bar games, revealing their differences from classical Nim and providing conditions for P-positions based on XOR operations.

Contribution

It characterizes P-positions in triangular chocolate bar games and highlights the differences in Grundy numbers compared to classical Nim.

Findings

P-positions satisfy x ⊕ y ⊕ z = 0

Grundy number of (x,y,z) is not always x ⊕ y ⊕ z

The structure differs from classical Nim

Abstract

Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rectangular chocolate bar game is a thinly disguised form of classical multi-heap Nim. In this work, we investigate the mathematical structure of triangular chocolate bar games in which the triangular chocolate bar can be cut in three directions. In the triangular chocolate bar game, a position is a $\mathcal{P}$-position if and only if $x \oplus y \oplus z = 0$, where the numbers $x,y,z$ stand for the maximum number of times that the chocolate bar can be cut in each direction. Moreover, the Grundy number of a position $(x,y,z)$ is not always equal to $x \oplus y \oplus z $, and a generic formula for Grundy numbers in not known. Therefore, the mathematical structure of triangular chocolate bar game is different from that of classical Nim.