Some sliding puzzles (for example see [Loyd]) have a close natural connection to monoids and permutations.
Solving Loyd's 15-16 puzzle (and its variants) seems to rely on stabilizer chains, with 3 generators for the submonoid of all words stabilizing the gap in its final position. The puzzle defined by a monoid representation is reduced to a permutation group element factorization problem in a low-degree representation of the Schützenberger group (see [Egn98] and [Ful96b] for the latter).
Besides stabilizer chains and similar stuff for Rubik game variants will be developed here, using CGT (computational group theory). According to [Hol05] no methods have been found for solving problems like finding the diameter of the Cayley graph of the Rubik cube group on its natural set of six generators (this is the same as the maximum number of quarter twists to restore the pristine state starting from an arbitrary position) that are significantly better than a brute-force depth first search. Well, perhaps the embedding into our morphing groups could give new insights (because they are more natural and continous in contrast to some general set of permutations)?!