This paper presents a study of chocolate bar games with a pass. Chocolate bar games are variants of the game Nim in which the goal is to leave your opponent with the single bitter part of the chocolate. The rectangular chocolate bar is a thinly disguised form of Nim. In this work, we investigate step chocolate bars of which the width is proportional to the distance from the bitter square. The mathematical structure of these step chocolate bar games is very di?erent from that of Nim. It is well-known that, in classical Nim, the introduction of the pass alters the underlying structure of the game, thereby increasing its complexity considerably; however, in the chocolate bar games treat in this paper the pass move is found to have a relatively minimal impact. Step chocolate bar games without a pass have simple formulas for Grundy numbers. This is not so after the introduction of a pass move, but they still have simple formulas for previous player’s positions. Therefore, the authors address a longstanding open question in combinatorial game theory, namely, the extent to which the introduction of a pass move into a game a?ects its behavior. The game we develop seems to be the first variant of Nim that is fully solvable when a pass is not allowed, and remains yet stable following the introduction of a pass move.
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