# Message #2135

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

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

- to the extent that you expressed concern that the

published numbers are in error. I remain exasperated that

you do not seem to be willing to embrace both the

permutations themselves as well as their equivalence

classes related by conjugation. BOTH are worth studying,

and it _starts_ with the distinct permutations from which

the equivalence classes can be derived. To take the

attitude that our apparent disagreement stems from some

personal focuses of interest is absurd. If you are so

focussed on certain equivalence classes, it would behoove

you to try to understand the issues in a manner better than

than your confused writings have indicated so far.

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

>swaps.)

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.

Regards,

David V.