Moves that do not commute: why order matters

Put on a sock, then a shoe. Now try it the other way round: shoe, then sock. The results are different. Cube moves behave the same way. Turn R and then U and you get one cube; turn U and then R and you get a different one.

Mathematicians say two moves commute if doing them in either order gives the same result. R and U do not commute, because they both move some of the same pieces: the corner and edge where the right and top faces meet. Opposite faces such as R and L share no pieces at all, so they do commute: R L and L R give the same cube.

Here is a second idea. Take any sequence of moves and repeat it again and again. Sooner or later the cube returns to where it started. The number of repeats needed is called the order of the sequence.

• R U R' U' has order 6: repeat it six times and the cube is back where it began.
• R U has order 105: you would have to repeat just those two moves 105 times.
• R2 L2 U2 D2 F2 B2 has order 2. One go makes a checkerboard pattern on every face, and a second go undoes it.

Why must the cube come back? There are only finitely many positions, and every move can be undone, so a repeated sequence has nowhere else to go but round in a loop that includes the start.

What about undoing a sequence? You might say “just do it backwards”. That is half right. To undo A B C you must reverse the order and invert each move: the undo is C' B' A'. Like undoing the sock and the shoe: first take off the shoe, then take off the sock.