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This question has been asked before on other sites, but none of the answers were satisfying to me.

I have heard people say that one can use group theory to solve rubik’s cubes. I understand why the set of rubik’s cube transformaions are a group.

But it’s unclear to me how using the conceptual apparatus of group theory helps you solve them.

E.g. the fact that many helpful operations are of commutator or conjugate form is claimed to entail that group theory is helpful here, but I discovered operations of this form in rubik’s cubes before I knew these more general terms, and it’s not clear to me how learning these terms adds to your understanding of Rubik’s cubes.

So what are concrete ways in which understanding group theory actually helps you in a non-trivial way to understand Rubik’s cubes and how to come up with solutions?

Mike Pierce
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user56834
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  • I think it is more accurate to say that being familiar with commutators and conjugates makes it easier to describe those key sequences of moves. As you said, you don't need a course in group theory to understand the use of such concepts in solving (or developing a method for solving) the cube. It just makes it easier to talk about it. I have described the use of both commutators and conjugates to a group of bright friends (but non-math majors) as well as to group of fellow grad students (back in the day). I simply used different language and terminology for the two target audiences. – Jyrki Lahtonen Aug 28 '18 at 05:07
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    In other words, Rubik's cube can be turned into a tool for teaching some elementary concepts in group theory. An extended example, if you like. You have probably seen better designed www-pages, but here's mine from the early 90s. Hope you are not afraid of a bit of Finnish. The pictures about the cube show commutators only, the idea of conjugates is in the text. – Jyrki Lahtonen Aug 28 '18 at 05:09
  • There are (many) algorithms describing how to solve this puzzle. In order zu show that these algorithms always work you need some parts of group theory. – Jens Schwaiger Aug 28 '18 at 06:18
  • @JensSchwaiger could you elaborate? – user56834 Aug 28 '18 at 07:34
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    One of the things group theory gives access to here is the study of shortest paths to a solution. In group theoretic terms, this is the question of largest distance between elements on the metric induced by choosing the basic moves as the set of generators. – Tobias Kildetoft Aug 28 '18 at 07:40

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