# Solve All Sequence (Rubik's Cube)

Can you prove/disprove that there is a solve-all sequence of moves to complete the Rubik's Cube from any solvable-position? If so, can you explain how long it is? If not, explain why not.

Just to be clear, I'm talking about a standard $3 \times 3 \times 3$ Rubik's Cube.

• Where do you find that problem? – YTS Feb 19 '15 at 2:44
• What do you mean by solve-all? If you mean a sequence of moves that you apply to any solvable cube to get a solved cube, of course not. Each move is a permutation $\sigma$. If $\sigma_n \circ \dots \circ \sigma_1$ is a "solve-all" sequence of moves for a solvable cube in state $\eta \neq \mathrm{Id}$, then $\sigma_n \circ \dots \circ \sigma_1 \circ \eta = \mathrm{Id}$, so $\sigma_n \circ \dots \circ \sigma_1 \circ \eta \circ \eta^{-1} = \eta^{-1} \neq \mathrm{Id}.$ – snar Feb 19 '15 at 2:44
• @snarski Would it be fair to say that, if there were such a move $\sigma$, then the Rubik's group would necessarily be cyclic? – pjs36 Feb 19 '15 at 2:55
• Or do you mean a sequence of moves which tours each possible position without repeating any. Thus if you apply the sequence you'll solve the cube at some point. Even if this is not a practical way to solve the cube. – Josh B. Feb 19 '15 at 2:56
• If I am reading it right, this question has already been answered on Puzzling SE. – George V. Williams Feb 19 '15 at 5:06

• Solve-all. ${}{}$ – warspyking Feb 20 '15 at 2:32