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A Rubick's Cube has owl heads on it, which can be misoriented. How many (times) MORE combinations are there of this cube vs. one that has blank stickers?

Can someone give me some hints? Thanks

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Rotating a face a quarter turn induces a four cycle on the eight corner cubies. Since a four cycle is odd the total rotation of all center cubies together must be an even number of quarter turns. I don't know if this is the only restriction on the possible configurations. –  WimC Apr 20 '13 at 20:36
    
Could you please explain in more detail what the difference is compared to an ordinary Rubik's cube? I am not quite sure what you mean by the owl heads.. –  Mårten W Apr 20 '13 at 20:41
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2 Answers

The answer is $4^6/2$ times more combinations than a regular Rubik's Cube. This is because the first 5 centers can be in any orientation, but the last center piece is restricted to only 2 possibilities instead of 4.

First notice that a quarter-turn is an ODD permutation:

$$(1234) = (41)(42)(43).$$

And since a solved (regular) cube is an EVEN permutation, that means any permutation that affects just the center pieces must also be even. So the MOST number of extra cube positions is $4^6/2$. Now we just need to show that all those positions are possible. It turns out they are.

By construction, it is possible to rotate a center piece 180 degrees:

$$(URLU^2R'L')^2.$$

This rotates the top center piece 180 degrees. To rotate the top center clockwise 90 degrees and the left center counterclockwise 90 degrees, do:

$$URL'F'BDU'L'UD'FB'R'L.$$

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And to round this off, the centre pieces are the only ones which are affected by the extra information, because the corners must be correctly rotated if the colours match. –  Peter Taylor Apr 20 '13 at 23:05
    
I have reasons to believe (4^6) /2 is waaay smaller than the correct answer. If the answer were true, it would mean [b]ONLY[/b] the centre pieces could rotate, which implies that once the cube is solved (ie., same colour pieces are on the same side) all the logos on the same side cube would always face the same direction, possibly except the centre piece, which may or may not face the same direction as the others. And that is just not true. I have a cube like that, I just messed up the orientations. –  One Two Three Feb 27 at 2:19
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Hint: Imagine fixing the center of the cube, and rotating only the six faces. The six center cubies (one in each face) do not move, only rotate. This determines the orientations of the eight non-center cubies on the top layer.

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