Superflip

The superflip or 12-flip is a Rubik's Cube configuration in which all 20 of the movable pieces are in the correct permutation, and the eight corners are correctly oriented, but all twelve of the edges are oriented incorrectly ("flipped"). It has been shown[1] that the shortest path between a solved cube and the Superflip position requires 20 moves under the usual half-turn metric (HTM)[lower-alpha 1], and that no position requires more moves (although, contrary to popular belief, there are many other positions that also require 20 moves).

computer graphic of the Superflip pattern

Under the more restrictive quarter-turn metric (QTM),[lower-alpha 2] the Superflip requires 24 moves,[2] and is not maximally distant from the solved state. Instead, when Superflip is composed with the "four-dot" or "four-spot" position, in which four faces have their centers exchanged with the centers on the opposite face, the resulting position may be unique in requiring 26 moves under QTM.[3]

Solutions

This is one possible sequence of moves to generate the Superflip (starting from a solved Rubik's cube), recorded in Singmaster notation; It consists of the minimal 20 moves under HTM,[lower-alpha 1] though it requires 28 quarter-turns:[lower-alpha 2]

One of the solutions using the minimal 24 quarter-turns[lower-alpha 2] is shown below (though it requires 22 HTM moves):[lower-alpha 1][4][5]

There is another solution that uses slice turns; it requires just 16 moves under the slice-turn metric (STM),[lower-alpha 3] (compared to 19 and 22 such moves in the first and second algorithm respectively), although there are 22 HTM moves[lower-alpha 1] and 32 quarter-turns:[lower-alpha 2]

Lastly, the following solution is both easiest to learn and fastest to do for humans, as the sequence of moves is very repetitive (even though it is not optimal under any metric – it requires 24 moves under STM[lower-alpha 3] and 36 moves under both HTM[lower-alpha 1] and QTM):[lower-alpha 2]

where y and z indicate rotations of the entire cube (and thus do not count as moves).

Properties

If you do a commutator with the superflip and any other algorithm, you will always end up back at the solved state; the superflip is also self-inverse, which means doing it twice will bring you back to the solved state. The superflip is completely symmetrical but not solved, so after applying the superflip to the cube you will allways get the same position, no matter how you're holding the cube. Any move will bring it to a position that is easier to solve.

See also

Notes
  1. Under HTM:
    * rotating a face either 90° or 180° counts as a single move;
    * a "slice turn", i.e. rotating a centre layer, counts as two separate moves (equivalent to rotating the two outer layers in the opposite direction).
  2. Under QTM, only 90° face-turns count as single moves; thus, a 180° turn counts as two separate moves, while a slice-turn counts as either two or four moves, depending on whether the slice is moved 90° or 180°.
  3. Under STM, 90° face-turns, 180° face-turns, and slice-turns (both 90° and 180° centre layer rotations) all count as single moves.

References
  1. Rokicki, Tomas. "God's Number is 20". Cube 20.
  2. The first algorithm is one of several 24 qtm solutions
  3. Rokicki, Tomas. "God's Number is 26 in the Quarter-Turn Metric". Cube 20.
  4. Joyner 2008, p.100
  5. Michael Reid (2005-05-24). "M-symmetric positions". Rubik's cube page. Archived from the original on 2015-07-06.

Further reading
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