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.

Figures (8)

• Figure 1: A move from ${\boldsymbol p} = (1,7,5,6,2,3,6)$ to ${\boldsymbol p}' = (0,1,5,4,2,3,6)$.
• Figure 2: A generic position in the game CN$(7,4)$, with $a = \min({\boldsymbol p})$.
• Figure 3: Visualization of the $\mathcal{P}$-positions of CN$(7,4)$. The sums of groups of stacks that are encircled are equal to each other or equal to the blue stack heights.
• Figure 4: Visualization of moves from $S_3$ to (a) $S_1\cup S_3$ (b) $S_4$.
• Figure 5: Generic positions with at least two zeros. (a) Two consecutive zeros. (b) Two zeros separated by one stack. (c) Two zeros separated by two stacks.

Theorems & Definitions (21)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: Theorem 1.2, tsF
• Theorem 2.1
• Definition 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• Definition 2.6