A group-labeling version of the Lights Out game

Parker, Darren B.; Zadorozhnyy, Vasily

doi:10.2140/involve.2021.14.541

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A group-labeling version of the Lights Out game

Darren B. Parker and Vasily Zadorozhnyy

Vol. 14 (2021), No. 4, 541–554

DOI: 10.2140/involve.2021.14.541

Abstract

The Lights Out game, originally an electronic game played on a $5 \times 5$ grid, is a solitaire game that can be played on any simple graph. It has been generalized to be played with multiple on-states, where each vertex state is represented by a label in $ℤ_{n}$ . We define a new variant of the Lights Out game, where the vertex labels can come from any group $H$ . This new game depends deeply on the group structure of $H$ and gives us a significantly different game for cyclic groups. We investigate problems related to counting winnable labelings in the case $H = ℤ_{n}$ .

Keywords

Lights Out, vertex labeling, games on graphs

Mathematical Subject Classification 2010

Primary: 05C20, 06B99

Milestones

Received: 6 November 2017

Revised: 19 June 2018

Accepted: 19 June 2020

Published: 23 October 2021

Communicated by Ronald Gould

Authors

Darren B. Parker
Department of Mathematics Grand Valley State University Allendale, MI United States

Vasily Zadorozhnyy
Department of Mathematics University of Kentucky Lexington, KY United States

Vol. 14, No. 4, 2021