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In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry.

What counts as a symmetry of the Rubik's cube? Is it a thing like, "turn the top face once clockwise, then once counterclockwise"?

How are these symmetries counted?

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Have you looked at this: en.wikipedia.org/wiki/Rubik's_cube_group ? –  Fredrik Meyer Mar 25 '11 at 18:29
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For what it's worth, that number doesn't match any of the various numbers given in the Wikipedia article en.wikipedia.org/wiki/Rubik%27s_cube. –  Grumpy Parsnip Mar 25 '11 at 18:30
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I was gonna write them all out, but the text box is too narrow to contain them. –  user641 Mar 25 '11 at 18:30
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I'm very curious to see where this number comes from. It is equal to $2^{41} * 3^{15} * 5^3 * 7^2 * 11 = (12!8!2^{12}3^8) 2^{12}$. I can see where the factor between brackets comes from, but the remaining $2^{12}$ is a mystery. –  Myself Mar 25 '11 at 18:46
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@Michael @Myself: This is the number of orientations of the central squares; see the edit in my answer below. –  joriki Mar 25 '11 at 18:51
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up vote 19 down vote accepted

Usually, something is called a "symmetry" if it leaves something invariant. For instance, the letter T has a symmetry in that you can take the mirror image with respect to its vertical axis and the shape remains the same.

In the case of the Rubik's cube, if you consider the coloured faces, there are no symmetries at all, since every move will change some of the faces. What is meant by "symmetries" in this case is operations that leave the "structure" of the cube invariant, i.e. that leave the cube in the same shape as before if you disregard the colours on the faces. These can be elementary operations, such as turning one face clockwise, or compound operations, such as the one you gave as an example, or more complex ones. The important point is that any sequences of operations that lead to the same end result are considered the same; for instance, if you turn a face clockwise three times, this is considered the same as turning it counter-clockwise once.

Since none of these operations leave the colours of the faces unchanged, and since they are considered different if and only if the end result is different, they can be counted by counting the number of different configurations into which they can bring the colours on the faces. So there's no need to think through all those gazillions of different sequences of moves; all you have to do is reason about which configurations of the faces are reachable through sequences of operations, and then count those.

Edit in reponse to the comment: Apologies for apparently giving an answer in the wrong "register"; it didn't seem from the phrasing of the question that you knew what a group was :-) Also apologies for not checking the numbers.

The number you cite is actually the total number of different positions of the cube pieces. Beyond the number of colour configurations reachable through turning faces, which is usually cited, this includes factors of $12$ for the number of different ways the pieces can be taken apart and put back together again, and $4^6$ for the number of different orientations of the central squares, which can't be distinguished from the colour markings. Of these $4^6=2^{12}$ different orientations of the central squares, $2^{11}$ can be reached without taking the cube apart. So denoting the number of configurations that's usually cited by $n$, you get

  • $n$ configurations without taking the cube apart and without marking the central squares
  • $2^{11}n$ configurations without taking apart but with marking
  • $12n$ with taking apart but without marking and
  • $2^{12}\cdot12n$ with taking apart and marking,

and the number you cited is that last number.

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@Joriki Thank you, but doesn't that give the wrong number? The size of the Rubik's cube group is 4*10^19 (en.wikipedia.org/wiki/Rubik's_cube_group). –  Mark Eichenlaub Mar 25 '11 at 18:36
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@Mark Eichenlaub: the Rubik's cube group only considers configurations that can actually be obtained by playing with the cube. This number is about the number of ways to assemble the cube, which is a somwhat larger number because some configurations cannot be obtained. (E.g. flipping a single edge.) –  Myself Mar 25 '11 at 18:53
    
@Joriki: Aha, I completely missed the indistinguishable $4^6$! I found it weird to think about them as real symmetries at first, but now I come to think about it, it makes a big difference if for instance you paint a picture on the cube. Thanks. –  Myself Mar 25 '11 at 18:55
    
@Myself @Mark: That distinction isn't quite right. According to the Wikipedia article (en.wikipedia.org/wiki/Rubik%27s_cube#Centre_faces), $2^{11}$ of the $2^{12}$ "invisible" orientations of the central squares are actually reachable without taking the cube apart, so the factors are actually $12$ if you allow taking it apart, $2^{11}$ if you mark the central squares and $12\times 2^{12}$ if you do both. –  joriki Mar 25 '11 at 18:57
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@Qiang Li: You're quite right. I guess this is a habit of speaking from physics, where by "symmetries" one usually means non-identity symmetries. –  joriki Mar 26 '11 at 9:45
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