Impartial and Partizan Restricted Chocolate Bar Games

TL;DR

This paper analyzes impartial and partizan restricted chocolate bar games, providing formulas for winning positions and revealing similarities in their game graphs, advancing understanding of combinatorial game strategies.

Contribution

It introduces formulas for winning positions in both game types and uncovers similarities in their game graphs, offering new insights into chocolate bar game strategies.

Findings

Formulas for winning positions for previous, Right, and Left players.

Similarity in game graphs between impartial and partizan versions.

Abstract

In this paper, we consider impartial and partizan restricted chocolate bar games. In impartial restricted chocolate bar games, players cut a chocolate bar into two pieces along any horizontal or vertical line and eat whichever piece is smaller. If the two pieces are the same size, a player can eat either one. In constrast, partizan restricted chocolate bar games include players designated as Left and Right and chocolate bars with black and white stripes. Left cuts the chocolate bar in two as above and eats the part with fewer black blocks. Similarly, Right cuts the bar and eats the part with fewer white blocks. A player loses when they cannot eat the remaining chocolate bar. We provide formulas that describe the winning positions of the previous player, Right, and Left players. We also present an interesting similarity in the graphs of previous players' winning positions for impartial and partizan chocolate bar games.