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Nitpick: aidos is right. The cube is always 20 moves away from a solution (but that solution typically isn’t the fastest possible (the 20 is in face turns, not quarter turns, so you can always add extra turns to the optimal solution to make the solution take 20 turns))

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That might very well be, but is it obviously so? What if there was a position that takes 19 moves to solve in the fastest way possible but the next-best solution takes 21 moves? Is there always a solution that takes an even number of steps?

Yes, there is. If the last turn is a quarter turn, replace it by a half turn and a quarter turn in the other direction. If it is a half turn, replace it by two quarter turns.

Well, okay. Technically you're right ;) But what if you count any movements that are followed by another movement of the same face as a single step?

The set of possible configurations would split in an even and an odd set, so there would be configurations that can’t be solved in God’s number of moves.

(AFAIK, we don’t know God’s number if we count moves that way. I would guess it’s larger than 20)

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