Hi!
I have a question about commutators. Can someone explain what the digits stand for when it for example look like this [4.3] or [3.1].

Andreas Andersson

Disclaimer: This might not be accurate, since I'm more familiar with commutators in their natural habitat (i.e. math) than with how they're used in puzzles, but in general, if you have two operations X and Y, the commutator of X and Y, written [X,Y], is the operation X Y X* Y*, where the * represents the inverse. For example, if X is UR and Y is F', then [X,Y] is the algorithm U R F' R' U' F. A surprising number of "nice" things you'd want to do on the cube can be expressed as a sequence of commutators. I suspect that the digits you see are just some abbreviation for a short algorithm, which you'd need to define; if there's a fixed canonical meaning for them, I don't know it and this post is useless. Say, if you were explaining a method, you might start by defining several short algorithms (or fingertricks), and then abbreviating later, longer algorithms as commutators and sequences of those building blocks.

There are lots of explanations of this commutator notation across many threads going back years.

In brief, [4,3] is short-hand to describe the general structure / length of a commutator. A commutator is a sequence of the form X Y X' Y' and the [4,3] short-hand says that X is 4 moves and Y is 3 moves.

The other notation we use is [N:M] where a ":" is used instead of a ",". This is for conjugate sequences (setup moves) and those sequences are of the form X Y X'. So [1:1] is a sequence where X and Y are both one move.

If you assume a [4,3] sequence is made up of nested commutators and conjugates then the only form it could be is [[1,1],[1:1]] (14 moves total).

I have seen your video "Understanding basic commutators and how they work" and it's give a really good explanation, but I haven't understood the digits before. Thanks for this explanation.
Andreas Andersson

Sure thing

We really should use something more technically correct like |X| = 4 and |Y| = 3 instead of [4,3] but the short-hand seems to work well enough.