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.

  • $\begingroup$ Where do you find that problem? $\endgroup$ – YTS Feb 19 '15 at 2:44
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    $\begingroup$ 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 $\sigma_n \circ \dots \circ \sigma_1 \circ \eta \circ \eta^{-1} = \eta^{-1} \neq \mathrm{Id}.$ $\endgroup$ – snar Feb 19 '15 at 2:44
  • $\begingroup$ @snarski Would it be fair to say that, if there were such a move $\sigma$, then the Rubik's group would necessarily be cyclic? $\endgroup$ – pjs36 Feb 19 '15 at 2:55
  • $\begingroup$ 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. $\endgroup$ – Josh B. Feb 19 '15 at 2:56
  • $\begingroup$ If I am reading it right, this question has already been answered on Puzzling SE. $\endgroup$ – George V. Williams Feb 19 '15 at 5:06

This website http://bruce.cubing.net/ham333/rubikhamiltonexplanation.html claims to have found a hamiltonian cycle for the Rubik's Cube group, which means that if you follow it, you will hit every possible position and end up where you started. Which means EVENTUALLY you will solve the cube, no matter where you start. Worst case scenario it will take the number of positions of the cube (43 quintillion) minus one moves lol.

  • $\begingroup$ Can you prove this is the shortest? $\endgroup$ – warspyking Feb 19 '15 at 9:29
  • $\begingroup$ What do you mean shortest? Shortest solution to solve a given position? Or shortest solve-all sequence of moves? $\endgroup$ – timidpueo Feb 19 '15 at 19:18
  • $\begingroup$ Solve-all. ${}{}$ $\endgroup$ – warspyking Feb 20 '15 at 2:32
  • $\begingroup$ It is the shortest sequence of moves because it's hamiltonian, which means it hits every position exactly once, never wasting a move by never repeating a position. $\endgroup$ – timidpueo Feb 21 '15 at 14:29
  • $\begingroup$ A simple sequence of 4 moves repeated can cycle through over 1000 positions. $\endgroup$ – warspyking Feb 23 '15 at 3:07

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