While the Hamiltonian circuit can apparently represent the various combinations, I considered that if we can start with a solved Rubik's cube and start making twists and turns and record the 'path' created in the Hamiltonian circuit by each twist of the cube and if all possible such paths are created, then would it be possible to mathematically estimate for any unsolved Rubik's cube, how close it is to getting solved? Because whatever unsolved state it is at, will be some position in the circuit path, and it should be possible to estimate a 'distance' or the number of twists required to reach a solution.
Would it be possible to make such an estimate?
UPDATE: There seems to be a clue here.