Sliding Block Puzzles with a Twist: On Segerman's 15+4 Puzzle

TL;DR

This paper generalizes Wilson's classification of sliding block puzzle solution groups to include puzzles with rotating tiles, revealing that solution groups are typically the entire generalized symmetric group or specific subgroups of index two.

Contribution

It extends Wilson's 1974 classification to puzzles with rotating tiles, identifying the structure of their solution groups as subgroups of generalized symmetric groups.

Findings

Solution groups are usually the entire generalized symmetric group.

Two exceptional cases have smaller solution groups.

Most puzzles' solution groups are either full or of index two.

Abstract

Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which the tiles rotate as they slide around. In 1974, Wilson classified the groups of solutions to sliding block puzzles. We generalize Wilson's result to puzzles like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are subgroups of the generalized symmetric groups. Aside from two exceptional cases, we see that the group of solutions to such a puzzle is always either the entire generalized symmetric group or one of two special subgroups of index two.