# Message #3402

From: Sid Brown <synjanbrown@gmail.com>

Subject: Re: [MC4D] Re: Cube puzzles and math

Date: Tue, 28 Jun 2016 08:42:26 +0100

Andrew,

Yeah I understand that you are talking about it quite abstractly. But what

I aim to do is program an algorithm that can always determine the shortest

possible solution. I can implement my brute force one for a 3x3x3 as that

won’t take too long, but for 3^4 it would take too long. I will use the

brute force of 3^3 to verify an algorithm of reduction of 3^3 and then

attempt to expand this to 3^4, but what I’m trying to do might prove near

impossible and I’m not sure whether I will actually achieve it. I will have

a decent stab at it though.

Sid

On 28 June 2016 at 08:36, apturner@mit.edu [4D_Cubing] <

4D_Cubing@yahoogroups.com> wrote:

>

>

> 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

>

>

>

>