“…However, it is clear that there are other elements in the groupoid which do not change the state (in the sense that the empty box is still in its place) but the pieces around it have been exchanged. For instance, if the original state is s 16 , consider the sequence of moves (taking into account the numbered boxes) (12,16), (11,12), (15,11), (16,15), where the pair (j, i) means that the move takes the box in position j (in the original configuration of boxes) to position i, that is, the move (15,16) will move the box sitting in position 15 down to occupy position 16, and so on. This would imply that the state s i is transformed in the state s j as the empty box changes from position i to j.…”