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If a cube is in a configuration that requires 20 moves to solve, is that sequence unique, or are there multiple sequences that arrive at a solution? That is: are there are two or more sequences that only have the start and finish position in common?

user3111005
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  • define "sequence"? Can I do "rotate this layer, unrotate, rotate, unrotate" to create a new sequence? – Guy Mar 13 '14 at 16:03
  • A sequence is 20 moves resulting in a solved cube. – user3111005 Mar 13 '14 at 16:10
  • Okay, exactly 20? – Guy Mar 13 '14 at 16:11
  • I would say $20$ or less. Otherwise, consider a cube with a solution of 18 moves, and start by doing/undoing anything gives several different solutions already. – mookid Mar 13 '14 at 16:16
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    Since there are more sequences of $20$ moves than there are positions of the cube, many of these will be non-unique. However, whether there are positions whose shortest solution is unique is another matter. Consider a cube where a single move has been made. There is only one way to restore it in one move. – Mark Bennet Mar 13 '14 at 16:30
  • @MarkBennet's comment I feel is the most elegant answer so far... proof w/o construction! – timidpueo Mar 13 '14 at 17:34

2 Answers2

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Of course there can be multiple possible solutions.

Before it was proved (via computer search) that every position required 20 moves or less, the Superflip was shown to require exactly 20 moves to solve. (See also here.) One sequence of moves which solves the Superfilp is

U R2 F B R B2 R U2 L B2 R U' D' R2 F R' L B2 U2 F2

However, the Superflip is symmetric in both rotations of the cube and in reflections of the cube. Therefore, one can modify the above sequence of moves to solve the position in a variety of other ways. For example:

U' L2 F' B' L' B2 L' U2 R' B2 L' U D L2 F' L R' B2 U2 F2 (reflection)
R D2 F B D B2 D R2 U B2 D R' L' D2 F D' U B2 R2 F2       (rotation)

Also, the Superflip when executed twice, results in the solved cube. So the reverse sequence works also:

F2 U2 B2 L' R F' R2 D U R' B2 L' U2 R' B2 R' B' F' R2 U' (reverse)

In summary, employing various kinds of symmetry you can easily generate multiple sequences for solving the same position.

Notes

  • Reflecting the cube across a plane parallel to faces R and L interchanges U with U', D with D', F with F', B with B', R with L', and L' with R'. This was used to generate the reflection sequence.

  • Rotating the cube with axis perpendicular to F and B is just a cycle on the faces R, U, L, D.

  • Reversing a sequence of moves is done by writing the sequence in reverse order and interchanging clockwise (no ') with counterclockwise (').

  • I have verified by computer that the above sequences of moves all generate / solve the Superflip.

Caleb Stanford
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Of course: consider symetry, for example:

a symetric pattern http://math.cos.ucf.edu//~reid/Rubik/Images/pons_asinorum.gif

solution!

mookid
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  • I should give you the benefit of doubt, and assume that you have checked this is solvable, but still I have to ask. Have you? All cube's aren't solvable. – Guy Mar 13 '14 at 16:12
  • the solution is here :) – mookid Mar 13 '14 at 16:14
  • okay that satisfies me. +1 :D – Guy Mar 13 '14 at 16:15
  • The most simple example is the first one, of course. I made the switch. – mookid Mar 13 '14 at 16:15
  • Sorry I don't understand how solving patterns has any bearing on my question. I have been told that the cube can always be solved in 20 moves or less. My question is: is that 20 move solution unique? My conjecture is: no its not unique. – user3111005 Mar 13 '14 at 16:21
  • @user3111005 mookid says that since this symmetric cube can be solved, you can mirror one solution to create a new solution. He proved your conjecture. Not unique. His link proves that the cube can in fact be solved. – Guy Mar 13 '14 at 16:22
  • This site: http//:www.cube20.org describes the "20 moves required" positions (they need "God's algorithm" apparently). It doesn't mention alternatives. Maybe alternative sequences are not possible. I feel I'm homing in on an answer... – user3111005 Mar 13 '14 at 16:37