Lights Out On A Random Graph
Bradley Forrest, Nicole Manno

TL;DR
This paper investigates the probability that a randomly chosen graph with n vertices results in a universally solvable Lights Out game, providing exact probabilities for small n and approximations for larger n using Monte Carlo simulations.
Contribution
It offers the first comprehensive analysis of Lights Out solvability probabilities on random graphs, including exact calculations for small graphs and Monte Carlo estimates for larger ones.
Findings
Exact solvability probabilities for n ≤ 11
Monte Carlo approximations for n up to 100
Analysis of connected graphs' solvability probabilities
Abstract
We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer , what is the probability that a graph chosen uniformly at random from the set of graphs with vertices yields a universally solvable game of Lights Out? When , we compute this probability exactly by determining if the game is universally solvable for each graph with vertices. We approximate this probability for each positive integer with by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs.
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Taxonomy
TopicsProbability and Statistical Research · Artificial Intelligence in Games · Game Theory and Applications
