MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but
- I happen to be a member of the Rubik’s Cube club here, and
- Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.
Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?
Annoying Precision 
Here is one result about “commutators tend to have fixed points” that I’ve heard in conjunction with solving the Rubik’s cube:
If a and b are two permutations such that every point bar one is a fixed point of either a or b, then the commutator of a and b is a 3-cycle.
Have you seen these posts on Rubik’s Cube?
I’m pretty sure I read through those and forgot about them. Thanks for the link!
When I first learned to solve the Rubik’s cube, the word of advice I received was “conjugate commutators”.
I can think of two reasons why your should expect commutators to do the trick. One is that commutators are reasonably likely to be in the kernel of quotients (especially when the quotients are abelian!). The other is that when you solve the Rubik’s cube, many of the “moves” aren’t just commutators, but commutators of relatively “basic” moves, which are largely disjoint in the parts of the cube they permute. But neither of these is a proof.
“The other is that when you solve the Rubik’s cube, many of the ‘moves’ aren’t just commutators, but commutators of relatively ‘basic’ moves, which are largely disjoint in the parts of the cube they permute.”
This. I don’t think the mathematical properties of commutators are as important as the fact that with commutators, it’s easy to construct something which is guaranteed to leave a large set of points — which you have a large degree of freedom in choosing — fixed.