Multi-player Small Nim with Passes

In Guy and Nowakowski's Unsolved Problems in Combinatorial Games, the following entry is found: "David Gale would like to see an analysis of Nim played with the option of a single pass by either of the players, which may be made at any time up to the penultimate move. It may not be made at the end of the game. Once a player has passed, the game is as in ordinary Nim. The game ends when all heaps have vanished." 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. (C) 2017 Elsevier B.V. All rights reserved.