On the Nature and Complexity of an Impartial Two-Player Variant of the Game Lights-Out

TL;DR

A recursive algorithm is utilized to compute the Nimbers of this variant of the solitaire game Lights-Out, where the player's goal is to turn off a grid of lights.

Abstract

In this paper we study a variant of the solitaire game Lights-Out, where the player's goal is to turn off a grid of lights. This variant is a two-player impartial game where the goal is to make the final valid move. This version is playable on any simple graph where each node is given an assignment of either a 0 (representing a light that is off) or 1 (representing a light that is on). We focus on finding the Nimbers of this game on grid graphs and generalized Petersen graphs. We utilize a recursive algorithm to compute the Nimbers for 2 x n grid graphs and for some generalized Petersen graphs.

Figures (17)

• Figure 1: An example of an initial state with $\Delta(G) \leq 2$ and $V_{0}^{(0)}(G) \neq \emptyset$.
• Figure 2: An example of an initial state with $\Delta(G) \leq 3$ and $V_{0}^{(0)}(G) = \{u_3\}$.
• Figure 3: A labeling of the vertices of $\mathcal{L}_{2,9}$ (top) and $P(9,1)$.
• Figure 4: $\mathcal{H}_9$ with $V^{(0)}_0=\{v_{0,1},v_{0,9},v_{1,1},v_{1,2},v_{1,8},v_{1,9}\}$.
• Figure 5: $\mathcal{D}_9$ with $V^{(0)}_0=\{v_{0,1},v_{0,2},v_{0,9},v_{1,1},v_{1,8},v_{1,9}\}$.

Theorems & Definitions (57)

• Definition 2.1
• Remark 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Corollary 2.5
• Proposition 2.6
• proof
• Remark 2.7
• Proposition 3.1