Message #3390

From: Sid Brown <synjanbrown@gmail.com>
Subject: Re: [MC4D] Cube puzzles and math
Date: Sun, 26 Jun 2016 15:48:56 +0100

Hi,

It is possible to define a rubik’s cube configuration as a non-commutative
ring (see https://en.wikipedia.org/wiki/Noncommutative_ring).
ring R. a,b are elements of the ring R. a,b are possible moves. a*b is the
move a followed by the move b. => a*b != b*a therefore non-commutative. 0
is the ID for the operation * and is therefore a solved cube. a fully
scrambled cube is the product of many moves and may not be in the simplest
form. a*a*a*a=a^4=0.
Storing it as a set of moves is an alternative to the type of group Joel
mentions which is storing the position of each piece and constraining the
movements. In the construct I describe the constraints are applied in the
definition of the possible elements of the ring R.

parity is formed as a move earlier in the algorithm has a side effect of
rotating 2 pieces when performed and later on the move that places these
pieces can only rotate 3. 3 modulo 2 != 0.

I hope this is easy enough to understand. It is probably worth researching
group theory to be able to fully understand this.

Regards
Sid

On 26 June 2016 at 14:22, e.kennedy.a@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Hello everyone!
>
> There has been a question on my mind for some time and after solving the
> 3x3x3x3 recently I getting to know that this group existed I thought that I
> could manage to find the answer here.
>
> I would like to know what’s the connection between maths and puzzles like
> the rubik’s cube. What specific areas of mathematics explain how rubik’s
> cube dinamics work?
>
> What do you think?
>
>
> P.S.: I was also wondering why does parity also exist in a 4x4x4 which
> orientable center pieces like the 4x4x4 axis cube.
>
>
> Thank you and best wishes!
>
>
>