A variation on the Rubik's cube

TL;DR

This paper generalizes the Rubik's cube puzzle to any 3-valent map, establishing an upper bound on the size of the associated group and conjecturing its tightness.

Contribution

It introduces a new mathematical framework for generalized Rubik's cube puzzles on 3-valent maps and provides an upper bound on their group sizes.

Findings

Established an upper bound on the group size for generalized Rubik's cube puzzles.

Conjectured that the upper bound is tight for all cases.

Extended the mathematical understanding of puzzle groups beyond the classic cube.

Abstract

The Rubik's cube is a famous puzzle in which faces can be moved and the corresponding movement operations define a group. We consider here a generalization to any $3$-valent map. We prove an upper bound on the size of the corresponding group which we conjecture to be tight.