As a puzzle, it can be a considerable challenge to return the tiles to their initial position once they have been scrambled. From a theoretical standpoint, it is interesting to consider which groups may be generated by these tile-sliding permutations. With one exception, the graphs included in this Demonstration all generate the full alternating group; all even permutations of the tiles can be achieved. The exception is the 4-prism, or cube graph, which generates a group isomorphic to the projective special linear PSL(2,7), the simple group of order 168. That means that if the tiles were placed randomly on the cube, there is only a 1/60 chance that the game could be solved by legal moves!

The 4-vertex graph of the tetrahedron (not included) is also "exceptional" (in that it fails to generate the full alternating group), but it is not very interesting. An unproven conjecture is that the cube and the tetrahedron are the only exceptions, and all other 3-regular graphs generate a full alternating group.