Multi-player Small Nim with Passes

This paper investigates the n-person combinatorial game of “Small Nim with Passes”, a variant of Nim, where players must always remove objects from the smallest nonempty pile and are allowed to “pass” their turn for a finite number of times. Let N be the number of piles in the game. When the number of players is greater than N+1, we determine all game values for all possible positions. The game values are determined completely when the number of players is equal to N+1. We also analyze certain cases of positions when the number of players is smaller than N+1, and leave some open problems that could be of interest to future research.