Rubik cube number of alternative solutions

Question

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?

Answer 1

Of course: consider symetry, for example:

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

solution!

Answer 2

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.