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Rubik’s Cube Not a Group?

I take a Rubik's cube in the solved state, and I secretly assign a unique integer label to each of the cubies. I then, via an arbitrarily long series of random moves, scramble and resolve the cube without looking at the integer labels. Will the integer labels be mapped back to their original positions? Does this answer change for a 4x4x4 cube?

If not, how large is the group of states for this integer labeled cube? Surely it must be quite a bit larger than the Rubik's cube permutation group?

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Yellow
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    I don't understand the question. When you scramble and re-solve the cube, all the cubies go back to their original position. Since you assigned the integers to the cubies, it seems that the integer must also go back to their original position. Can you clarify? – Ted Jan 02 '13 at 20:20
  • @Ted I'm trying to understand if the solved state is unique. – Yellow Jan 02 '13 at 20:21
  • How could it not be unique? There's only one solved state, right? I still don't understand. – Ted Jan 02 '13 at 20:22
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    In the 3x3x3 case the answer is yes, the solved state is unique. The corners and edges have unique color combinations, leaving only the center square on each face, which is unique in having only the one color. Indeed the face-centered squares are sometimes considered immovable. – hardmath Jan 02 '13 at 20:24
  • @Ted There are 54 cubie faces, right? While solving a cube we only consider 6 colors, where each color maps to one of nine possible integer labels. Why is it true that the integers won't be scrambled in the solved state? – Yellow Jan 02 '13 at 20:24
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    The cubies on a 3x3 cube have "home positions" based on the centres of each side which remain in the same relative positions throughout. The only variation in the solved cube is the possible rotation of the centre cubies from their original state - identifying this requires asymmetric markings on the centres. – Mark Bennet Jan 02 '13 at 20:25
  • @Mark Bennet Ah, so there are multiple possible solution states, four per center cubie? – Yellow Jan 02 '13 at 20:28
  • @Mark Bennet Does this extend to other cube states, and thus, increase the Rubik's cube permutation group by some factor $C$? – Yellow Jan 02 '13 at 20:28
  • Are these $4^6$ states all reachable? – Yellow Jan 02 '13 at 20:31
  • @MarkBennet And if I understand correctly then, a 4x4 cube would not have "home" positions for each of the cubies based on edges? – Yellow Jan 02 '13 at 20:37
  • In a 4x4x4 (or nxnxn) cube consider the corner cubies - they determine the orientation of the centres. I do think this is all available online, though, with some simple searching. – Mark Bennet Jan 02 '13 at 20:44
  • @MarkBennet What I've found online seems to claim that it is not the case that the orientation of the cubie centers is fixed? – Yellow Jan 02 '13 at 20:52
  • What is unclear to me is whether all of these orientations are reachable, increasing the total number of cube states by some $C=4^6$. – Yellow Jan 02 '13 at 20:53
  • I have done quite a bit of searching, but I've been finding conflicting information about whether the orientations of the cubie centers are uniquely determined. It seems to me they should not be. – Yellow Jan 02 '13 at 21:04

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Based on the comments, it seems like the question is whether the orientation of the center squares is fixed. The Wikipedia page for Rubik's cube says that only half the center orientations ($4^6/2$) are achievable.

Here are instructions for 2 types of center rotations: (1) rotate one center piece clockwise and an adjacent center piece counterclockwise; (2) rotate a single center by 180 degrees. (Corners and edges stay where they are.) Together these generate a subgroup of index 2 inside $(\mathbb{Z}/4\mathbb{Z})^6$ (the subgroup where the sum of all entries is even).

Ted
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