Given a black box that can quickly calculate the number of moves from solved a given Rubik's cube is, we can imagine puzzles and games for humans:
Given two cubes, which is closer to solved?
What move from a given state gets closer to solved?
Players take turns trying to make a cube closer or farther from solved. Parity in the Quarter Turn Metric might be useful or interesting here.
God's Algorithm is practical for 2x2.
Very similar to chess endgames.
EmoticonEmoticon