Restricted Nim with a Pass

This paper presents a study of restricted Nim with a pass. In the restricted Nim considered in this study, two players take turns and remove stones from the piles. In each turn, when the number of stones is $m$, each player is allowed to remove at least one stone and at most $\lceil \frac{m}{2} \rceil$ stones from a pile of $m$ stones. The standard rules of the game are modified to allow a one-time pass, that is, a pass move that may be used at most once in the game and not from a terminal position. Once either player has used a pass, it is no longer available. In classical Nim, the introduction of the pass alters the underlying structure of the game, significantly increasing its complexity.<br> In the restricted Nim considered in this study, the pass move had minimal impact. There is a simple relationship between the Grundy numbers of restricted Nim and the Grundy numbers of restricted Nim with a pass, where the number of piles can be any natural number. Therefore, the authors address a longstanding open question in combinatorial game theory:<br> the extent to which the introduction of a pass into a game affects its behavior. The game we developed is the first variant of Nim that is fully solvable when a pass is not allowed and remains fully solvable following the introduction of a pass move.

Paper

Similar papers

© 2026 NYSGPT2525 LLC