You may want to e-mail this person as to what computer solver they used for their video.
Sorry, but what do you mean by commutators?
According to my search, commutators are just notations of basic operations.
As for how commutators relate to cubing we generally say they are permutations of the form ABA'B' where both A and B are permutations in and of themselves.
A good example:
A = M
B = E
ABA'B' becomes:
M E M' E' which is a commutators that achieves a patter with the centers. If you use commutators where the B part is a single slice or face turn, and the A part is a 3 move conjugate maneuver (XYX') then you can achieve some very powerful, and useful effects on the cube.
A good example of this would be:
A: R' D R
B: U
ABA'B' becomes:
R' D R U R' D' R' U' which is a very quick, easy, and intuitive corner 3 cycle for blindfolded solving.
Any solution to a cube can be represented as a seqence of commutators, isn't it?
No, this is not true. However, it is
very close to being true. To be very precise:
Any legal state of an n x n x n cube may be solved using only commutators as long as the parity of all piece orbits is even.
Basically commutators only solve even permutations. So, any scrambled state of any cube can be solved by first figuring out the parity state of the wing edges, corners, and central edges (if they exist). Next you will apply up to, but no more than, n/2 quarter turns of the necessary layers for an even cube or (n-1)/2 quarter turns of the necessary layers for an odd cube. It is possible that you don't need to apply any quarter turns to affect parity. After doing this you will have restored all piece obits to even parity, and the cube can now be solved using commutators.
Commutators, in my opinion, would be a terribly inefficient algorithm for solving the general n x n x n cube on a computer. Basically you would need very close to 9 moves on average to solve 2 pieces.
There are 6n^2 - 12n + 8 pieces to the n x n x n cube, and some of them are likely to be solved. However, for the sake of argument consider that they are all unsolved. This means it will take you approximately:
(9/2) * (6n^2 - 12n + 8) = 27n^2 - 54n + 36 turns to solve the cube
Keep in mind that I am ignoring the parity fix turns at the beginning, as this is only an estimate. This would result in approximately 264000 turns to solve a 100x100x100 cube using commutators. Or for a better example approximately 250 turns to solve a 4x4x4 when computer algorithms, and even people (!), are capable of solving it in less than 100 turns.
Chris
