Message #2135

From: David Vanderschel <>
Subject: Re: [MC4D] Calculating the number of permutation of 2by2by2by2by2 (2^5)
Date: Thu, 10 May 2012 22:11:26 -0500

I accidentally composed my last message in this thread
with a line length that was too long for the way I have
set up my email program for plain text. Thus it really
trashed the formatting to an extent that I imagine a lot
of folks would not want to read. Thus I am reposting
the message with a line length that will render
properly. Sorry about the first mess.

Melinda wrote:
> I’m sorry that I have exasperated you. I really don’t
> mean to argue anything and I believe that I get what you
> are saying. I think the tension is due to the fact that
> each of us is interested in related but different
> concepts here. In the 2D case you see 24 distinct states
> in a way that is both mathematically clean, meaningful
> and interesting to you. I on the other hand see only 8
> states that are interesting to me, however messy they may
> be to count.

I am interested both in the number of distinct states as
determined by the positions and orientations of the cubies
as well as the equivalence classes of those states under
conjugation by a symmetry. Your "interest" in the
equivalence classes under conjugation (or whatever you have
in mind) seems to go so far that you would actually exclude
the distinct states on which the published counts are based

It occurred to me earlier today that it may well be that
you lack adequate grasp of the group theory involved - as I
cannot imagine that, if you really understood the concept
of conjugation in the context of permutations of a finite
set, you would remain as confused as you apparently have.
I urge you to study a little group theory as it relates to
permutations. In particular, learn about representing a
permutation in terms of its disjoint cycles, what cycle
structure is, and why permutations with the same cycle
structure must be conjugates of one another. This is not
deep stuff. It is basic group theory. You should not be
sticking your head in the sand every time somebody says
"conjugation"! And please note that there are definitely
others besides myself who have tried to point you in the
direction of conjugation. I had merely been a little more
patient about it - up to now.

> Regarding recolorings, please don’t think that I care
> about the particular colors. I only care about the
> patterns that they create. Perhaps we can clear this up
> by talking about those color pairs that are important to
> you. Imagine a pristine cube. If you swap two opposite
> sets of stickers, then you feel that nothing has changed
> even though you can’t rotate the whole cube to match the
> original colors.

The symmetry in that case is reflection in the zero plane
of the axis on which you swapped the sticker colors.

>(Perhaps you mean to allow only an even number of pair

No. Swapping just one opposing pair is OK.

>For the things that I care about, definitely nothing has
>changed when you do that because the puzzle is still in
>what I consider to be *the* solved state which I define as
>the one in which all sides have a single different color.

You seem to be losing sight of the fact that you want to
use color remapping on different piles in permuted states
to make them match up. I agree that doing it on a single
pile that is in initial state does not mean much. But,
when you do it on piles, which started out with the same
color scheme but which have been permuted in distinct ways,
with the purpose of relating those permutations, you have
to go about it in such a way that you do not create two
different puzzles. I.e., both piles must still contain the
same set of cubies as determined by their sticker colors.
Going back to the basic Rubik example, if you exchanged the
blue and yellow colors without also swapping white and
green, then the edge cubie that had been white/blue would
become white/yellow, and the puzzle is not supposed to
contain any edge cubies colored that way. It is no longer
meaningful to compare permutations between puzzles that
differ in such a profound way.

>For that matter I am also perfectly happy swapping the
>stickers of two adjacent faces because that gives the same
>result as when swapping opposite sides. The puzzle remains
>solved. Unless I’ve just said something whacky, I think
>we can leave it at that.

Unfortunately, you have indeed said something wacky because
what you have described does not relate in any valid way to
the method which I think you propose to define your notion
of distinct states. (I have to be vague about this because
you have never actually spelled your method out
unambiguously. I have been trying to help you make it
meaningful by urging you to add the opposing pair
restriction, which addition you resist. Nevertheless,
adding that restriction makes it equivalent to conjugation
by a symmetry, which is on a sound basis and does not even
require talking about colors.) It’s wacky either because
it is trivial (and not worth saying in the first place) or
because it is meaningless for whatever purpose you would
seek to employ it.

Earlier in this thread I wrote:

If you really believe that you are driving at some sort<br>
of equivalence that goes beyond conjugation by a<br>
symmetry, then I believe the burden is on you to spell<br>
it out in much greater detail, identify its utility,<br>
and make it meaningful in terms of the 'mechanics' of<br>
the puzzles.

You have not made any credible attempt to do so.
Furthermore - and to put it bluntly - I don’t believe that
you can.

David V.