Message #590

From: David Smith <>
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, 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, "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
> 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
> that in MC4D we should be allowed to do only a move that will be a
> of two rotations (where when saying rotation I mean a 90º

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,