Orders of moves in the Rubik's cube

Playing with my Rubik's cube, I was thinking of facts about it that are immediate to mathematicians but novel to others. Here's one:

Given a Rubik's cube in the solved state, any sequence of moves, if repeated long enough, will eventually return the cube back to the solved state

To a mathematician, that follows from "The moves of a Rubik's cube form a finite group. Therefore all the elements have finite order."

Each atomic move of Rubik's cube has order 4. The simplest compound move is R U (rotate the right face 90° clockwise, then rotate the top face 90° clockwise). What's the order of R U?

Secondly, what's the greatest order of an element in the Rubik's cube group?

I tried repeating R U for a long while. I lost count (~50 repetitions), but it did eventually return home.

• According to Wikipedia, the greatest order of an element is $1260$. – mjqxxxx Aug 20 '12 at 19:18
• $\operatorname{ord}(\mathbf{RU})=105$ by actual manipulation. – Brian M. Scott Aug 20 '12 at 19:31

Take a solved cube and do $RU$; then trace the cycle structure of the permutation it realizes.

The combination moves 5 corner cubies in a cycle where a cubie is twisted by a third of a turn when it gets back to its original position, so that's a factor of 15.

It also twists a the FRU corner by one third of a turn; that's taken care of by the factor of 15 too.

Then it permutes 7 edges cyclically, but this time every edge has the correct orientation when it gets back.

So the order is the least common multiple of 7 and 15, namely 105.

(For a subercube we need another factor of 4 to get the centers back into the original orientation).

For some time ago I wrote my bachelor thesis about the element of greatest order:

http://www.math.kth.se/~boij/kandexjobbVT11/Material/rubikscube.pdf

Actually, I came up with a new theorem for the generalized symmetric group that confines, as a special case, the orders inside the Rubik's cube and the result is quite interesting. The greatest order is 1260.

• How long is r u or r u r' u for that matter? – N3buchadnezzar Aug 25 '14 at 17:25