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.
Best Answer
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).