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 starting position, any sequence of moves, if repeated long enough, will eventually return the cube to the starting position
(The moves of a Rubik's cube form a group, a subgroup of some finite permutation 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.