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I am interested in knowing (a little more) about the mathematics underlying some of Erno Rubik's games. I guess we all know his famous cube. I heard at some point that a solution was possible because it has the structure of a group. Is it true?
There is a similar game by Rubik, not so famous and easier to solve, consisting of a cylinder. I would like to know about the mathematical structure behind it. Is it also a group? Which is the reason for its being much easier than the cube, given that the parameters (colours, number of slots or positions) are, I believe, the same?

Thanks in advance

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  • $\begingroup$ It is true that the moves you can make on Rubik's cube (and on his other puzzles) form a group, and knowing about that group can help you solve puzzles. My guess is that the kind of information you are after is available on the web by means of a simple search. $\endgroup$ Oct 26, 2015 at 8:15
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    $\begingroup$ For the first paragraph, some other posts tagged rubiks-cube+group-theory might be of interest for you. $\endgroup$ Oct 26, 2015 at 8:48
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    $\begingroup$ See math.harvard.edu/~jjchen/docs/… $\endgroup$ Oct 26, 2015 at 9:31

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The main reason why group theory is useful for solving Rubik's cube is that the set of available moves is the same in each state of the cube.

There are other twisty puzzles where some moves are possible only in some state -- these are often referred to as "jumbling" puzzles by puzzle aficionados. As a simple example, imagine taking an ordinary Rubik's cube and gluing two of the little cubies in the solved state together, such that moves that take them apart are now disallowed. This resolves in a puzzle with fewer states than the original cube, which can nevertheless be harder to get a grip on because you can't mix and match combinations as freely as you usually can. And group theory is not particularly helpful for expressing this restriction.

(Actually, gluing one pair of cubies together is not so bad; you can keep that pair fixed in space and twist the cube around them, still yielding a group. And practically, many solution methods can fairly easily be adapted to deal with a single glued pair. Do it to two pairs of cubies, and trouble will begin to pile up, though).


I don't think "consisting of a cylinder" pinpoints the particular puzzle you're thinking about well enough to say something about it in particular. There are many different more-or-less cylindrical derivatives and variants of the cube.

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  • $\begingroup$ As far as I know, the fact that the cube's moves group has a lot of small normal subgroups is quite helpful, too, because this gives many easily composable moves. $\endgroup$
    – A.P.
    Oct 26, 2015 at 11:06
  • $\begingroup$ @Henning Makholm. Thanks for your answer and nice to see you here again...By the way, I tried to send you a paper on ideals some weeks ago, but it returned as unknown email.....If you want, you can open a chat window, and we exchange the data. Best regards $\endgroup$ Oct 26, 2015 at 12:34
  • $\begingroup$ @Henning Makholm. I have had a new look at the objects at stake and, other than the number of elements per side (some variant have the same, some others not), a critical fact seems to be that the cylinder presents an empty spot to play around in the rotation (which allows to dislocate each bit), whereas the cube obviously does not allow that. I guess that is why the cylinder was always way too easier to me than the cube.......No, is it still a group with that empty slot? $\endgroup$ Oct 26, 2015 at 13:26

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