It might be helpful to note that the moves form a group, but the puzzle itself is the group action of this group on the cubes/squares/pieces. In this context i

It might be helpful to note that the moves form a group, but the puzzle itself is the group action of this group on the cubes/squares/pieces. In this context it reveals the usefulness of commutators, which are moves of the form ABA'B', where A and B are some sequence of moves and ' denotes inverse. In general A and B aren't commutative, but with respect to a subset of the pieces, the actions of A and B are effectively commutative, so ABA'B' will do nothing to those pieces. Which means you can get some simple permutation control over the remaining (usually much smaller) pieces of the puzzle.