Message #1845

From: Melinda Green <>
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 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?