# Message #592

From: David Smith <djs314djs314@yahoo.com>

Subject: Re: [MC4D] Something interesting and strange about permutations

Date: Sat, 27 Sep 2008 13:35:31 -0000

Hi David,

— In 4D_Cubing@yahoogroups.com, David Vanderschel <DvdS@…> wrote:

>

> On Friday, September 26, "David Smith" <djs314djs314@…> wrote:

>

> >— In 4D_Cubing@yahoogroups.com, David Vanderschel <DvdS@> wrote:

>

> >> David Smith, you might want to consider extending your

> >> permutation counting exercise to include also puzzles

> >> with mirroring allowed. …

>

> >Yes, I will definitely work on this new problem. By the

> >way, from my point of view the super-supercube, while being

> >a generalization of the regular cube, is the most elegant

> >and simplest form of the Rubik’s Cube. In d dimensions,

> >it’s a hypercube subdivided into n^d hypercubies, each of

> >which is uniquely identifiable in any position or orientation.

> >Any 1 x n^(d-1) group of hypercubies can rotate freely

> >around the point at the center of the group of hypercubies

> >in a manner which brings each hypercubie’s position to

> >a position previously occupied by a hypercubie.

>

> Actually, this is how I have always viewed the order-3

> puzzles (any dimension). And, yes, the view applies

> as well to puzzles of order higher than 3. What you

> are describing as "1 x n^(d-1) group of hypercubies"

> is what I call a "slice". I distinguish between

> external slices and internal slices.

Yes, I also refer to these as slices!

> My lack of interest stemmed from what I perceived as a

> lack of practical physical realizablity, since the

> group permutes what amount to stickers that can never

> be visible. But I must agree that it can be simulated

> in an understandable way.

I like to think of the super-supercube as a mathematical

entity, rather than a physical reality. I don’t really

think of the hypercubies as having stickers at all. They

are just uniquely identifiable in any position or

orientation.

> >If we include reflections, they can occur about any

> >(n-1)-dimensional space which contains the point at

> >the center of the group of hypercubies and is

> >orthogonal to the faces of the hypercubies it

> >intersects.

>

> You can also reflect about hyperplanes that are on

> diagonals. The reflection planes must be such that

> the puzzle transforms onto itself, but that does not

> mean they must be axis-aligned. E.g., for the

> 3-puzzle, a plane of reflection can intersect the

> locations of 4 corners in such a way that 2 pairs are

> diagonal from each other across opposing faces and 2

> other pairs are adjacent along opposing edges. (The

> last 2 pairs are what I call "triagonal" from one

> another, diagonal within the 4-point rectangle.)

Yes, I forgot about these reflections. I suppose I was

thinking about the fact that any reflection can be produced

by the type of reflections I mentioned.

> >> >MC2D moves only do interesting things with odd

> >> >numbers of reflections,

>

> >> ? I do not understand the above statement. Why is a

> >> 4-cycle (Rotate corner positions.) not interesting?

> >> [(1,2)(3,4)] * [(2,4)] = [(4,3,2,1)]

>

> >You are definitely correct here David, but your example

> >is unfortunately not (it actually contains 3

> >reflections).

>

> Perhaps we are not together on the permutation

> notation. I am using cycle representation. If you

> want to think about it geometrically, assign indices

> to the corner positions in clockwise order starting at

> the upper left hand corner. Then [(1,2)(3,4)] is a

> reflection about the y-axis, [(2,4)] is a reflection

> about the NW->SE diagonal, and [(4,3,2,1)] is a

> 4-cycle rotating all four corners counter clockwise.

>

> >An even number of reflections can only produce an even

> >permutation of the cubies,

>

> Not true. Reflections can be odd or even and there is

> one of each in the two I composed above. (Fairly

> obvious, with 2 2-cycles in one and only 1 2-cycle in

> the other.) So the product, a 4-cycle, is odd.

> Furthermore, the product is not mirroring. (It is

> true that the composition of two reflecting

> permutations is always a non-reflecting

> reorientation (or identity if the same reflection is

> used twice).)

>

> >and an odd number of reflections, an odd

> >permutation. (For example, reflecting two adjacent

> >sides of MC2D would create a 3-cycle of the corners,

> >which is not possible with an odd number of

> >reflections. Of course, you wrote a very lengthy

> >reply which was very accurate, so this small error is

> >understandable! :)

>

> I suppose yours is somewhat larger since you implied

> that I had erred when I had not; but I will forgive

> you anyway! And please do not hesitate to attempt to

> correct me if, in future, it again looks as if I have

> erred. Next time it may be so, and I would not wish

> to go uncorrected. I do err frequently because I

> often have complex thoughts. I learn by being

> corrected, and I occasionally assert something I am

> not quite sure about with the hope that someone will

> point me in the right direction if I turn out to be

> confused.

I am quite familiar with cycle notation! I believe

you are making an error as to what a move is in MC2D.

You seem to be talking about reflecting the entire

square! As in MC3D, a move is any rotation or

reflection of a face that preserves the general shape

(i.e. takes positions of cubies to positions of

previous cubies). I believe you are thinking about

the entire puzzle as a face of MC3D, because the moves

you are describing are reflections of a face in that

puzzle.

If you launch MC2D, it will be immediately clear that

there is only one move, and that is to swap two

adjacent corners. Thus, (1,2)(3,4) is actually two

moves, one reflection of the North face and one of

the South face. If you were to actually reflect

the entire puzzle, the centers would also reflect,

and thus would just be a reorientation of the puzzle,

and not a move at all.

> After I posted my previous message it occurred to me

> that I missed yet another possibility and an even

> larger group. We might call it a SUPER–super-

> supercube. I had written,

>

> >If you try to flip a slice of the 3-puzzle in 3D, you

> >do wind up (uselessly) with it being outside in; but

> >it is not reflected.

>

> I now regret that "(uselessly)" parenthesis. Let us

> take the attitude that an order-m n-puzzle is an m^n

> stack of nD cubies. (My n is your d and my m is your

> n. I prefer to stick with n for the dimension of the

> puzzle.) Also let us imagine that _every_ one of the

> 2n facelets on each cubie has a sticker. Use a normal

> scheme for assigning colored stickers to the visible

> facelets. I am willing to make all the invisible

> stickers be black; but it would not be a necessity.

> In the order-3 cases, it is certainly OK as each cubie

> is uniquely identified by the combination of colored

> stickers it bears. Now I want to say that, in

> addition to our familiar twists, you can remove a

> slice, flip it over with respect to the axis in which

> it is ‘flat’, and replace it. (I can imagine no real

> mechanism, but you can easily simulate the 3D cases

> with a pile of cubical blocks. Just a little tedious

> to reorient the whole cube or to remove a slice and

> replace it flipped - but certainly possible.) Now

> this is not mirror reflection; but it is a new kind of

> permutation. The colored stickers can be turned to

> face inwards and have valid possible internal

> positions. Flipping of slices need not be limited to

> external slices.

>

> Imagine that all 2n*m^n facelet positions are indexed

> relative to fixed spatial coordinates and we identify

> the stickers on them in the initial position by the

> position they occupy. The permitted alterations to

> the pile can be seen as permutations of the 2n*m^n

> stickers. There must be some sense in which the

> familiar group of the regular order-m n-puzzle is

> still in there as a subgroup; but I am not sure yet

> how to characterize that, since there are extra

> stickers. (May require a quotient.) We now have the

> ability to turn over any slice relative to the

> dimension in which it has thickness 1. When the

> puzzle is scrambled, many initially-visible colored

> stickers may no longer be visible; but they can still

> be accessed by appropriate manipulation.

>

> Anyone for slice-swapping? ;-)

A very interesting idea, and congrats for thinking

of it! Who knows, I may end up doing my permutation

formulas for these puzzles as well! :)

All the Best,

David