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It's been well known for a few years now that every Rubik's Cube scramble can be solved in 20 moves.

However, this is only for the case where the orientation of the centers is not taken into consideration. I couldn't find a reference for the picture cube, where face orientation does matter. Does anybody know a resource on the minimum number of moves required to solve a picture cube?

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  • $\begingroup$ Not a complete answer, but even on a standard rubiks cube, the sides and corners must be in exactly the correct position and orientation. The only difference is that each of the centers may be rotated to one of four orientations. Hopefully that is a start for those who write an answer. $\endgroup$
    – Cort Ammon
    Commented Jan 24, 2015 at 5:59
  • $\begingroup$ I think a counting argument gives a lower bound of 22. For a trivial upper bound, is there a short sequence to rotate a single center cube or multiple of them? $\endgroup$
    – xnor
    Commented Jan 24, 2015 at 9:45
  • $\begingroup$ @xnor: Could you expand on your counting argument? It's not obvious to me. $\endgroup$
    – BmyGuest
    Commented Jan 24, 2015 at 13:55
  • $\begingroup$ @xnor Counting only gives you 21. $\endgroup$
    – Jakube
    Commented Jan 24, 2015 at 22:00
  • $\begingroup$ By "minimum," do you mean "maximum" or "least upper bound"? The minimum number of moves required to solve a cube is $0$, which occurs when every piece is already in the correct position. $\endgroup$
    – KSmarts
    Commented Jan 26, 2015 at 19:34

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If you mean "minimum" as a lowerbound, then yes. Last year there was a detailed discussion on the 3x3x3 "supercube" (which is a different name to describe the same puzzle).

Note that they found that God's number is 26 quarter turn moves last year for the regular 6 color Rubik's cube as well.

You will see in this discussion that a lowerbound of 24 half turns and 28 quarter turns has been established.

http://cubezzz.dyndns.org/drupal/?q=node/view/536

For your possible curiosity, Bruce Norskog recently found it that it takes at most 12 half turns to solve the first 4 cross edges and correctly twist the 5 centers adjacent to them.

https://www.speedsolving.com/forum/showthread.php?50422-Optimal-supercube-cross

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