# 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!

>

>

>