# Message #1844

From: David Vanderschel <DvdS@Austin.RR.com>

Subject: [MC4D] Re: State graph of MC2D

Date: Wed, 03 Aug 2011 19:20:35 -0500

Andrew Gould wrote:

>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.

But the context in which it is "practical" is an exhaustive

analysis of all positions. It is practical in the sense

that it pares down the number of arrangements that have to

be analyzed. That is not what is going on in the MC2D case.

The argument there involves a graph relating particular

equivalence classes of permutations, apparently with the

claim that this graph helps you solve the puzzle from a

particular state. Different problem.

>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.

I agree absolutely, and it makes perfectly good sense. As

Nan has commented, he had failed to mention that, in

addition to the rotational symmetry, reflections were also

admitted. I erred when I claimed that allowing

conjugating with [the symmetry corresponding to a reflection

about a diagonal axis] would produce the larger full

conjugacy classes. It does not. (Had I not made that

mistake, I might have made it all the way to realizing that

the equivalence relation for Melinda’s partition is

"conjugates by a symmetry".)

Now I am trying to figure out why my weird way of stating

the equivalence relation works relative to this much more

sane symmetries approach.

Regards,

David V.