The algebraic solvability of the novel lamplighter puzzle

Article type: Research Article

Authors: Gregor, Connor^{; *} | Ashlock, Daniel | Willms, Allan R.

Affiliations: Math & Stats Department, University of Guelph, ONT, Canada

Correspondence: [*] Corresponding author. E-mail: cgregor@uoguelph.ca.

Abstract: In this study, the group of finite cyclic lamplighter states is reinterpreted as the novel lamplighter puzzle. The rules of the puzzle are outlined and related back to properties of the lamplighter group with specific interest placed upon the discussion of which puzzle instances are solvable. The paper shows that, through the use of algebra, many puzzle instances can be identified as solvable without the use of an exhaustive search algorithm. Solvability depends upon the creation of irregular generating sets for subgroups of the finite cyclic lamplighter group and the cosets formed by these subgroups. Further possible generalizations of the lamplighter puzzle are also discussed in closing.

Keywords: Lamplighter group, generating subgroups, theoretical solvability, novel game design

DOI: 10.3233/ICG-210195

Journal: ICGA Journal, vol. 43, no. 4, pp. 203-225, 2021