# Question on symmetry of the Rubik's cube



## Robert-Y (Mar 20, 2009)

I attended a lecture yesterday in London. Marcus du Sautoy (who some of you may know, is a reasonably famous mathematician) took the lecture. He posed a question to the whole audience about the Rubik's cube. I can't really remember what the exact question was but it was something like "How many symmetrical positions are there of a Rubik's cube?" (I think he was talking about rotational and refletion symmetry although I might be wrong). Anyway, the answer contained 24 or 25 digits and I was totally confused by the answer. Could someone figure out and explain how this number was obtained? (Btw, don't think too hard, the solution is simple, as I remember he didn't write a lot for the solution)


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## qqwref (Mar 20, 2009)

I don't really understand the question, can you be more precise? Like, what's a "symmetrical position" - is it a position that looks the same when you do a rotation of some kind, or is it a position that could look the same if you execute it on two cubes with different initial orientations, or what? And how can the answer be 24 or 25 digits if there are only 20 digits' worth of positions in the cube to start with? Questions, questions, questions...


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## pinoycuber (Mar 20, 2009)

well.. cube is on geometry right?
did he solve it shapes ?


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## Robert-Y (Mar 20, 2009)

Oh yeah, you treat the colours of the faces in a special way but I don't really know how, either he didn't explain the question well enough or I wasn't paying enough attention. 

I think when he was speaking about symmetrical position, he was talking about:

1. any position (including illegal positions) which can be rotated round to get a similar position e.g. popping an edge piece, twisting it and putting it back will give 2 symmetrical position if you rotated the cube round the edge piece which is flipped.

2. any position which has a line(s) of symmetry.

3. perhaps other types of symmetry I don't know about.

"And how can the answer be 24 or 25 digits if there are only 20 digits' worth of positions in the cube to start with?" This is the main reason why I asked this question. Unfortunately, I was forced to leave asap, because we were running slightly late, and we would have missed dinner, so I didn't get the chance for to ask him your exact question. I'm almost certain that he's including more than just all of the legal cube positions.


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## Lucas Garron (Mar 20, 2009)

http://www.geocities.com/jaapsch/puzzles/symmetr1.htm


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## Robert-Y (Mar 20, 2009)

Thanks a lot Lucas, now I'm not annoyed and frustrated at the answer


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## cuBerBruce (Mar 21, 2009)

The number of legal symmetric positions is given here.


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## Robert-Y (Mar 21, 2009)

Cool, my guess was about 1trillion for legal symmetric positions which is what I thought the guy meant in the first place. Btw, the person who got closest to the answer won the challenge and Marcus named a "group" after the guy who got the closest answer.

I didn't like the question at all because I didn't know the answer included all illegal positions as well. I might have gotten close to the answer if I didn't know a lot about Rubik's cubes lol (i.e. if I didn't know there are 43 quintillion legal cube positions).


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## Robert-Y (Apr 27, 2009)

Is there some simple formula to working out the number of symmetrical positions? I think there is...

If there is one, can someone please explain it briefly?


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## Robert-Y (Apr 27, 2009)

12! x 2^12 x 8! x 3^8 x 4^6

(Edge permutation) x (Edge orientation) x (Corner permutation) x (Corner orientation) x (Centre rotation?)

Is this correct?

It's really just a guess.


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## spdcbr (Apr 28, 2009)

I know 1 symmetry! U2 D2 L2 R2 B2 F2
yay! should we write all the combinations of symmetries down?


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## miniGOINGS (Apr 28, 2009)

spdcbr said:


> I know 1 symmetry! U2 D2 L2 R2 B2 F2
> yay! should we write all the combinations of symmetries down?



AKA: E2 M2 C2  lol


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## Stefan (Apr 28, 2009)

Robert-Y said:


> Is there some simple formula to working out the number of symmetrical positions? I think there is...


I don't.



Robert-Y said:


> 12! x 2^12 x 8! x 3^8 x 4^6
> Is this correct?


Not at all.

Do you know why they're called "symmetric"?


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## miniGOINGS (Apr 28, 2009)

StefanPochmann said:


> Robert-Y said:
> 
> 
> > Is there some simple formula to working out the number of symmetrical positions? I think there is...
> ...



lol are you just saying "no" or are you having a facepalm moment?


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## Stefan (Apr 28, 2009)

miniGOINGS said:


> lol are you just saying "no" or are you having a facepalm moment?


No, this is relatively hard stuff and theoretical, too, so I already respect him for being curious about it. And I suspect he just doesn't know what's meant with "symmetric", hence the question/suggestionForResearch I added in the meantime.


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## Robert-Y (Apr 28, 2009)

The thing is Marcus (the person who took the lecture) asked us to work it out, and after the correct answer was revealed, he quickly explained how to do it and it took him less than a minute to explain. I think he could be wrong although the chances are, he's probably not.

Does anyone else think that the answer does not contain 24, 25 or 26 digits?


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## Stefan (Apr 28, 2009)

Like Bruce mentioned earlier, the number of legal symmetric states is only 164,604,041,664. Twelve digits, not 24 or 25. Far less than what you're getting. States are called symmetric iff they have some kind of symmetry, and most states simply have no symmetry at all.

The usual Tperm has two-fold symmetry - you can mirror it front<->back and still have essentially the same case (just with a different color scheme). Similarly, MEM'E' can be rotated three ways around the UFL-DBR diagonal to reach essentially the same state, and you can mirror it to double that for six-fold symmetry. The A-perm however has no symmetry, just like most states. The solved state and superflip are the symmetry kings, having 48-fold symmetry. This all was for when you consider rotating and mirroring the cube for symmetry. Inverting can add another factor 2 (that way the G1-perm is symmetric to the G3-perm, if you know what I mean).

At least I think so, I must admit I'm not 100% comfortable with these things myself. Which is part of why I very much doubt anyone can sufficiently explain this in a minute to non-experts. And I think for determining the number of symmetries, there's no simple formula but it's an elaborate task of covering all those different symmetries and how many states each of them has. Very messy. I'd love to be shown wrong, but I doubt there's something simple.

Probably that guy didn't mean symmetry the way we mean it, or you misunderstood something.


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## qqwref (Apr 28, 2009)

Yeah, if you didn't understand it enough to remember his explanation, you might well have messed up the question too  If it was that many digits, and he explained it that way, it can't have been the symmetries - maybe it was just the positions on a normal cube, or something like that?


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## Robert-Y (Apr 28, 2009)

http://www.city.ac.uk/news/archive/2009/03_March/25032009_1.html

I'm trying to find when he asks the question in the video.

EDIT: Go to around 54:50


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## qqwref (Apr 28, 2009)

Yep, as expected, he's definitely using a completely different definition of "symmetry" than what cube theorists use. I really don't like his approach to name a group after some random student, though. That's more reminiscent of "You're the millionth visitor!" type advertisements than of science.

The only thing I can think of is that he's considering the group of all positions a Rubik's Cube can possibly be in - allowing for positions which cannot be accessed using only turns, and counting any possible rotations. That's 1.24 * 10^22... so 23 digits. I don't think it's even possible to get a figure more than that unless you start moving center caps or stickers around (or unless you have a supercube).


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## Robert-Y (Apr 28, 2009)

Ok after watching the last 5 minutes or so, I realised I was wrong, he just gives you a few hints on how to get to the actual answer, the answer isn't that simple.

So, he thinks that the number is around 2 x 10^24. Hmm....

Anyone want to try and obtain this number somehow?


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## spdcbr (Apr 28, 2009)

I don't like lectures...was it a good lecture? I hope you didn't get in seirous trouble.


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## MistArts (Apr 28, 2009)

Robert-Y said:


> Ok after watching the last 5 minutes or so, I realised I was wrong, he just gives you a few , the answer isn't that simple.
> 
> So, he thinks that the number is around 2 x 10^24. Hmm....
> 
> Anyone want to try and obtain this number somehow?



(8!) x (3^8) x (12!) x (2^12) x (4^6) = 2.12592246 × 10^24

Including center orientation...


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## cuBerBruce (Apr 29, 2009)

I watched the video, and I believe I understand it now. The number he is asking about is the number of symmetries of Rubik's cube. This is not the number of symmetrical positions.

He mentions the number started with a 2, followed by 24 digits. So I'm pretty sure the number Robert mentioned is the right number. But I believe it is arrived at using |G|*2048*24. Here |G| means the size of the standard Rubik's cube group (<U,D,L,R,F,B>) which we know is 43252003274489856000. It's multiplied by 2048 because it's possible for the 6 centers to be rotated that many combination of ways for a given element of G. In other words, |G|*2048 is the number of positions for the 3x3x3 supercube. Finally, it is multiplied by 24 because that is the number of ways the Rubik's cube can be reoriented.

Now to explain more why I think that is the right answer.

In the lecture, he talks about the symmetries of two objects with six symmetries, an equilateral triangle, and a starfish (one with 6 arms apparently). The triangle can be rotated 3 ways in its plane to leave it looking the same, or it can be reflected (or rotated so you view it from its other side) or not reflected for another factor of 2 symmetries. The starfish being a phyiscal object can not be transformed into a reflected starfish, and if you rotate it to view it from its other side, it doesn't look the same. Similarly, the (geometrical) cube has 48 symmetries. We can reorient it 24 ways without changing its appearance, and we can reflect it without changing its appearance. (Because of its 3-dimensional nature, reflecting it is not the same as viewing it from the other side.)

So now the Rubik's cube. Basically, he asked something like "How many ways can we pick it up, manipulate it, and set it back down so that it looks the same as it did before it was picked up?" (Well, we have to consider the cube with its colored stickers removed to have it really look the same.) Without "cheating" (disassembling and reassembling the cube, for instance), we can move the corners and edges around |G| ways with respect to the centers. If we know how the corners and edges have been permuted, we know that there are 2048 combinations of ways the centers can be rotated. Since these are physically realizable changes we can make to the cube (and don't change the appearance of the cube), these count as symmetries. Finally, when we set the cube back down there are 24 different orientations we can place it down without affecting how the cube looks. (We can't, for instance, put it down rotated horizontally 45 degrees, because then it will appear to have moved from how we had it originally.) Like a starfish, we can't physically reflect it. We can rotate it to view it from the other side, but we've already counted that. So |G|*2048*24 is the number of ways we can manipulate it without changing how it appears.


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## spdcbr (Apr 29, 2009)

cuBerBruce said:


> I watched the video, and I believe I understand it now. The number he is asking about is the number of symmetries of Rubik's cube. This is not the number of symmetrical positions.
> 
> He mentions the number started with a 2, followed by 24 digits. So I'm pretty sure the number Robert mentioned is the right number. But I believe it is arrived at using |G|*2048*24. Here |G| means the size of the standard Rubik's cube group (<U,D,L,R,F,B>) which we know is 43252003274489856000. It's multiplied by 2048 because that's the number of it's possible for the 6 centers to be rotated that many combination of ways for a given element of G. In other words, |G|*2048 is the number of positions for the 3x3x3 supercube. Finally, it is multiplied by 24 because that is the number of ways the Rubik's cube can be reoriented.
> 
> ...


Ahhhh! Don't say complicated stuff like that unless you want to kill people. My head hurts. You're too smart for me.


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## byu (Apr 29, 2009)

spdcbr said:


> cuBerBruce said:
> 
> 
> > I watched the video, and I believe I understand it now. The number he is asking about is the number of symmetries of Rubik's cube. This is not the number of symmetrical positions.
> ...



If you don't want that to happen, just don't visit the Puzzle Theory section.


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## spdcbr (Apr 29, 2009)

Actually, I found this thread on the home page thank you very much.


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## qqwref (Apr 29, 2009)

spdcbr said:


> Ahhhh! Don't say complicated stuff like that unless you want to kill people. My head hurts. You're too smart for me.



Ahhhh! Don't say stupid stuff like that unless you want to offend people. My head hurts. You're too dumb for me.

(But yeah, don't go in a topic in the puzzle theory section if you don't want to hear complicated stuff.)


Nice analysis, Bruce


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## Stefan (May 1, 2009)

At 56:16 the guy also said "How many numbers does it have?" which just shows once again that he's just terrible.


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