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.

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


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.

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

The order of any alg (sequence of moves) can be calculated by checking the resulting cycles. For each cubie of the cube, follow its displacement. When it's back in place, determine if it is twisted or flipped. Then you take the smallest common multiple of all cycles. (A bit more care needs to be taken, but that's the general idea.)

For a quick calculation, you can use https://mzrg.com/rubik/ordercalc.shtml


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