Commutators

Every algorithm on this site was constructed by somebody. They did not find them by trial and error, and there is no book they came from. This lesson is the tool they used, and it is the one thing in the expert track that changes what the cube is rather than how fast you finish it.

The pattern

A commutator is written [A, B] and means: do A, do B, undo A, undo B. If A is R U R' then undoing it is R U' R', so [R U R', D] written out in full is R U R' D R U' R' D'.

If A and B had nothing to do with each other — say R and L — the four parts would cancel and the cube would be untouched. All the interest is in what happens when they overlap slightly.

Three is the smallest number

You cannot swap two pieces and leave the rest of the cube alone — you met that as a beginner, as the reason a single flipped edge means somebody has taken the cube apart. Any sequence that restores everything else must move pieces in threes, or in pairs of pairs. So the three-cycle is the atom of cube algorithms, and the commutator is the machine that makes them.

Interchange and insertion

In practice you build a commutator out of two named parts. The interchange is usually a single turn that carries one of your three pieces into the place of another. The insertion is a short sequence that reaches the third piece — and touches exactly one square the interchange also touches, which is the condition the whole thing rests on.

Take three corners: the two on the right of the top layer, and the one at the bottom-front-right underneath them.

The two halves each wreck the cube and each undo their own wreckage. The only place the damage cannot cancel is the up-front-right corner, which is the one square both halves touch — and that single overlap is what produces the cycle.

Edges, in four turns

Four turns for a three-cycle is close to the cheapest thing on the cube, and it is not a coincidence: the slice turn and the half turn of the top overlap in exactly one edge.

Setting the case up

A commutator only cycles the three pieces it happens to reach. When the pieces you want are somewhere else, you move them into reach first and put everything back afterwards — a conjugate, written [S: X] and meaning S, then X, then undo S.

Building one for a case you have never seen

Here is the situation this pays for. You finish the first two layers and find a corner in the wrong slot — two corners in the bottom layer and one in the top all need to move round each other. No algorithm sheet covers it, because it is not a case anyone has named.

1. Name the three pieces and where each has to end up. Say it out loud; the whole thing falls apart if you are vague about the cycle.
2. Find two of the three that a single turn swaps. That turn is your interchange. If no single turn does it, add a setup turn until one does.
3. Find a short sequence that carries the third piece into the place the interchange acts on, without touching the other two. Three turns is the usual length: out, across, back.
   R^'D^'R
4. Write it in the pattern — insertion, interchange, insertion undone, interchange undone — and run it on a solved cube first.
   R^'D^'RUR^'DRU^'
5. If the pieces land in the right places but facing the wrong way, keep the interchange and change the insertion. There is nearly always another route down.

Commutators hiding in your algorithms

Once you can see the pattern you will find it everywhere. The A permutation is a three-cycle of corners and can be written as two commutators stitched together. The E permutation is four corners in two swaps, and is built the same way. The T and Y permutations are a corner swap and an edge swap held together by a conjugate.

Some things are not commutators, and it is as useful to know which. Sune twists three corners and cycles three edges: it is not a three-cycle of anything, so no single commutator produces it. Changing orientation without moving pieces takes a pair of commutators working against each other, which is why the algorithms that do it are long.

This is the point where the cube stops being a list of cases. You will not stop using memorised algorithms — they are faster, and speed is a real thing to want — but you will stop believing that a case you have never seen is a case you cannot solve.