Impartial and Partizan Restricted Chocolate Bar Games

TL;DR

The work analyzes impartial and partizan restricted chocolate bar games within combinatorial game theory, deriving explicit Grundy-number formulas for Maximum Nim with $f(m)=\lfloor m/2\rfloor$ and presenting a complete, closed characterization of $\mathcal{P}$-positions for the associated 2D chocolate-bar game via a 2-adic decomposition of $x+1$ and $y+1$. Extending to the partizan setting, the authors classify positions into $\mathcal{P},\mathcal{N},\mathcal{L},\mathcal{R}$ through explicit families, illustrating rich structural parallels and differences with the impartial case and providing a 3D generalization framework. They also connect their results to historical work (IOI2005) and discuss the implications of 2D results for higher-dimensional generalizations, while acknowledging the substantial challenges in obtaining full formulas for all previous-player positions in the partizan model. The study thus advances explicit, constructive descriptions of winning positions in both impartial and partizan restricted chocolate bar games and outlines clear avenues for future 3D investigations.

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.

Figures (19)

• Figure 1: Original chocolate bar with a bitter part
• Figure 2: Traditional Nim with two piles
• Figure 3: Chocolate bar with a bitter part, where some squares are removed
• Figure 4: Chocolate bar without any bitter part
• Figure 5: (5,4)

Theorems & Definitions (41)

• Theorem 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Definition 3.1
• Remark 2
• Theorem 3
• proof