Message #3401

From: apturner@mit.edu
Subject: Re: [MC4D] Re: Cube puzzles and math
Date: Tue, 28 Jun 2016 00:36:20 -0700

Sid,

You’re talking about actually finding an explicit presentation of the Rubik’s Cube group. Just from some quick Googling, it looks like there has been a fair amount of discussion of this online, but no definitive results as far as I can see.

But in terms of just talking about the group, it isn’t necessary to actually determine the presentation. We still understand what it means to say that we quotient out the free group by the equivalence relation that relates physically equivalent states.

Another way to think about it, which sort of bridges the gap between your formulation and Joel’s, is given on Wikipedia. They just consider the 48 movable stickers on the cube, and the permutations of these stickers that correspond to the moves L, R, U, D, F, B, and then say that the Rubik’s Cube group is the subset of the symmetric group S_{48} generated by those permutations: < L, R, U, D, F, B>. This phrases the setup in terms of moves of the cube, but automatically deals with the physically equivalent states, because it inherits the relations from the presentation of S_{48}. In fact, this approach gives you a presentation of the Rubik’s Cube group, in the form < L, R, U, D, F, B| all relations of S_{48}>. This presentation is almost certainly not the smallest, however!

Cheers,
Andrew