The Rubik cube is an example of both a 1-person game, and a game for which all positions can be enumerated. The number of moves taken to restore a cube to its 'home position' could be called the 'depth' of that position, analogous with nomenclature in Awari, Checkers and Chess. There can be more than one metric: in chess, more than one goal can be identified - mate, mate and/or conversion of force, mate, mate and/or conversion of force, mate and/or conversion of force and/or Pawn-push. Thus, the metric defines the graph and the depth. The key is to define an indexing of the positions, i.e., a 1-1 into mapping of {positions} --> Z, preferably [0, N] with N as small as possible. The indexing method should be usable to get the index or to derive the position of the cube from the index. Still, I would have thought that 'N' is still challenging for today's computers. 7-man chess endgames, involving an N of about 60^7 are on the limit at the moment. G