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## Nationals 2010

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?

### 5 Responses

1. 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.

2. Have you seen these posts on Rubik’s Cube?