Circular Nim CN(7,4)

Abstract

Circular Nim is a two-player impartial combinatorial game consisting of $n$ stacks of tokens placed in a circle. A move consists of choosing $k$ consecutive stacks and taking at least one token from one or more of the stacks. The last player able to make a move wins. The question of interest is: Who can win from a given position if both players play optimally? In an impartial combinatorial game, there are only two types of positions. An $\mathcal{N}$-position is one from which the next player to move has a winning strategy. A $\mathcal{P}$-position is one from which the next player is bound to lose, no matter what moves s/he makes. Therefore, the question who wins is answered by identifying the $\mathcal{P}$-positions. We will prove results on the structure of the $\mathcal{P}$-positions for $n = 7$ and $k = 4$, extending known results for other games in this family. The interesting feature of the set of $\mathcal{P}$-positions of this game is that it splits into different subsets, unlike the structure for the known games in this family.