# Message #590

From: David Smith <djs314djs314@yahoo.com>

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

Date: Sat, 27 Sep 2008 03:29:28 -0000

Hello David, nice to meet you. I agree with your observations

very much about this topic.

— 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. I am actually curious about

> how many-fold is the increase even for the 3x3

> 3-puzzle. (For myself, the mirroring issue seems more

> interesting than the super-supercube considerations.)

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. 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.

> >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).

An even number of reflections can only produce an even

permutation of the cubies, 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! :)

— In 4D_Cubing@yahoogroups.com, "lucas_awad" <lucasawad@…> wrote:

> I want to speak about what I said about the special rotations that

> should only be allowed in MC4D.

> What I was thinking about was that if we say that a rotation is a

set

> of two reflections, and we are only allowed to do rotations in a 3D

> cube, and reflections in the MC2D, then it would be logical to

think

> that in MC4D we should be allowed to do only a move that will be a

set

> of two rotations (where when saying rotation I mean a 90º

rotation).

That’s an interesting way to view the analogy of allowable

moves in higher dimensions, Lucas. In my way of thinking,

I would not want to restrict the rotations that could be

done on MC4D, as all rotations are allowable (nothing about

4D space prevents this). MC2D can only perform reflections

due to the fact that at least 3 dimensions are required

for the type of rotation implied by the Rubik’s Cube (A

rotation that preserves the shape and position of the layer

of cubies while not breaking away from the rest of the cube).

Reflections however, require the cube to be in a space at

least one dimension higher than the cube itself for the

move to be physically possible. Therefore, when only

considering what is physically possible, when using the

cube in a space of dimension equal to the dimension of the

cube no moves can be performed on MC2D, while any rotation

that meets the restrictions above can be performed on a face

of a higher-dimensional cube. If using the cube in a space

of higher dimension than it, reflections can also be

performed (and hence the only moves allowable on MC2D).

Of course, one can always restrict what moves

can be performed on any permutation puzzle, so your idea

on what moves can be performed on MC4D is not incorrect

by any means; it is just a further restriction of all

possible moves, which results in a slightly different

puzzle (and which represents a subgroup of the group I was

considering, with a smaller order).

I am very close to getting the super-supercube formula.

I have been busy lately while also catching up on some

reading in my spare time, which is why I have not

finished yet (with the experience of figuring out

the other 3D and 4D formulas, it is not that time-

consuming). I will probably post the formula tommorow,

and I will then consider cubes with reflections allowed.

All the Best,

David