Message #1841

From: schuma <mananself@gmail.com>
Subject: Re: State graph of MC2D
Date: Wed, 03 Aug 2011 18:38:02 -0000

Andy,

I think this is the direct link to the page you were talking about:

http://kociemba.org/math/symmetries.htm

I think Kociemba’s definition is aligned with Melinda’s classification. In that webpage he explained the concept by two ways.

(1) In the first paragraph (box/example/whatever) he did use the word recolor. So it’s recolor + whole cube rotation/reflection.

(2) Then he gave his preferred definition in the form of B=S’*A*S. In the process of S’*A*S, the first step S’ is a whole-cube rotation/reflection of a solved cube. This step is nothing but recoloring with some constraints (like, white and yellow cannot be on adjacent faces etc). So the two interpretations are the same thing. The latter one is just more analytical.

I think Melinda’s equivalency for MC2D can also be defined in the analytical way like S’AS:

Let’s say state A can be obtained from the solved state by applying permutation A (with a little abuse of notation), and state B can be obtained by permutation B. As David said, A and B belong to S_4. Then A and B are equivalent if there exists S in {whole-puzzle rotation and/or reflection, i.e., dihedral group D_4} such that B=S’AS.

In my email last night I forgot to include reflection. Now I remember Melinda’s definition does need reflection for S.

Anyway this equivalency is weaker than conjugacy because S can only be a whole cube operation rather than a general twist. That’s why David found Melinda’s classes should be merged to get the conjugacy classes.

Nan

— In 4D_Cubing@yahoogroups.com, "Andrew Gould" <agould@…> wrote:
>
> oops, speaking of link problems, if you click the link, it doesn’t give you
> a left side. Copy & paste http://kociemba.org/cube.htm. Also, by
> ‘rotations’ and ‘reflections’ I mean applying them to the whole puzzle.
>
> –
> Andy
>
>
> From: 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] On Behalf
> Of Andrew Gould
> Sent: Wednesday, August 03, 2011 12:54
> To: 4D_Cubing@yahoogroups.com
> Subject: RE: [MC4D] Re: State graph of MC2D
>
> �
> Hi David,
>
> Herbert Kociemba (part of the cube20.org group) used this ‘reducing by
> symmetries’ to solve god’s number is 20. It cut the number of states they
> had to solve by a factor of about 39.7. So it’s quite practical. He didn’t
> call it ‘recoloring’ though….
>
> He defined a ‘symmetry’: a combination of rotations or a combination of
> rotations with a reflection. Each symmetry, S has a unique inverse S’, and
> the double inverse, S’’ = S. Two states, j and k are then said to be
> equivalent if there exists a symmetry such that j = SkS’. I look at it as
> conjugating by symmetries only.
>
> For details, go to http://kociemba.org/cube.htm, then on the left side under
> ‘The Mathematics behind Cube Explorer’, click ‘Equivalent Cubes and
> Symmetry’.
>
> –
> Andy
>