# Message #1845

From: Melinda Green <melinda@superliminal.com>

Subject: Re: [MC4D] Re: State graph of MC2D

Date: Wed, 03 Aug 2011 18:48:57 -0700

On 8/3/2011 5:20 PM, David Vanderschel wrote:

> 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.

I think that we are in some ways talking past each other, and I suspect

that where we split is in our overall goals in which the word

"practical" means very different things to each of us. Where I started

was in trying to understand the whole puzzle, which to me clearly meant

getting very familiar with all the possible states and how best to find

my way to the solved state. When I see a state that is one twist from

being solved, I don’t particularly care if the twisted face is on the

top, left, bottom or right. Also, I don’t particularly care about the

color of the face that needs to be twisted. I simply began to get to

know a bunch of the recurring patterns and wanted to enumerate them all

and see how they relate to each other.

You looked at my diagram and thought that what I was attempting had

something to do with cycles and conjugates but was just doing it wrong.

Different problems indeed. I don’t know the precise mathematical

language to express my diagram in your terms but it looks like you and

Nan are doing a good job of that.

I find it fascinating that the 3^3 can be reduced to a graph with a only

a million such nodes, if I understood Nan correctly. That says to me

that it should be possible to print the whole thing out on a single

wall-sized diagram! Maybe "practical" is too loaded a term. Certainly it

is a very compact form and turned out to be practical in that it lead to

a tractable computer proof of God’s algorithm. I understand that that as

applied math, this is a sort of practicality is of little interest to a

pure mathematician.

> […] (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.

Maybe this is fruitful because we are both interested in symmetries.

Where we part ways maybe has to do with our differing goals. I don’t

expect you to be interested in the same symmetries that I am. It’s

exciting to me to have a proof of God’s number but that doesn’t

necessarily give you a deeper understanding of the puzzle. I am happy

with the result but still a bit unsatisfied. I want to know on a deep

level why the number is what it is and how the internal states

interrelate. MC2D is a kind of work-up towards that. I never had much

trouble solving it, but I didn’t feel that I understood it. Since I

mapped it I feel that I completely grok God’s algorithm in this model

puzzle. With my diagram it is trivial to see that God’s number must be 3

whereas before then I wasn’t quite sure. I can now also see exactly how

the puzzle changes if we allow middle slice twists. Now I want to learn

about the equivalences and differences between the corresponding graphs

of all such puzzles. All I’m really asking is to know the mind of God.

Is that too much to ask?

-Melinda