# Message #2131

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

Subject: Re: [MC4D] Calculating the number of permutation of 2by2by2by2by2 (2^5)

Date: Thu, 10 May 2012 19:05:51 -0500

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