Date: Wed, 05 Jun 96 23:07:24 -0400
From: Jim Mahoney <mahoney@marlboro.edu >
Subject: Re: A essay on the NxNxN Cube : counting positions and solving it

>>>>> "Jerry" == Jerry Bryan <jbryan@pstcc.cc.tn.us> writes:
Jerry> This has been discussed before on Cube-Lovers, but I am
Jerry> still puzzled or curious about the usage of the word
Jerry> "orbit".

So am I, actually. Dan Hoey just replied with a much more detailed
understanding of the group theory aspects of this word than I have at
present, which I'll have to think about some more. For myself, I mean
no more and no less than a set of cubies which can move into each
other's positions. For a 3x3x3 cube which I imagine to be made of of
3^3=27 smaller cubes (cubies), what I call "orbits" are exactly those
cubies at the 8 corners, 12 edges, 6 faces, and 1 (unseen) at the
center.

Jerry> Secondly, if my understanding of your model is correct, you
Jerry> are treating positions as distinct which cannot be
Jerry> distinguished with normal coloring of a physical cube (even
Jerry> an imaginary physical cube for large N).

Yes, exactly. As Dan just said, he has discussed this vision of the
cube in earlier notes, and called it the "theoretical invisible cube".

When I started thinking about these larger cubes, I built them by
making piles of dice. All the inner cubies were there, and all had
definite orientations, and I could see them every time I tried to
rotate a slice - which required carefully seperating out the layers,
turning one, and putting everything back together. So perhaps that's
why I liked those "invisible" inside pieces. But it also seemed more
elegant. The restricted versions (only the outside, only the
orientations of the corners and edges, etc.) are all special cases.

Jerry> There are several implications of how you treat visibly
Jerry> indistinguishable positions. For example, it impacts your
Jerry> counts of how many positions there are. For another
Jerry> example, it impacts your solutions (e.g., "invisible"
Jerry> incorrect parity on the 4x4x4. "Invisible" bad parity can
Jerry> also occur on the 3x3x3 if you remove the face center color
Jerry> tabs. A slice move will give the edges and corners
Jerry> opposite parity that is not visible.) Perhaps you could
Jerry> discuss these issues with respect to your model.

I'm not sure what there is to say; you seem to understand the issues.

Yes, I am counting "visibly indistinguishable" positions as different,
especially on the larger cubes, if by "visibly" you mean to only look
at the outside. I'm assuming that either the whole thing is
transparent, or that you can take it apart, and see the inside cubies
if you like. There are parity constraints between the different
orbits, including the ones on the inside that are "invisible," but
they turn out to be fairly simple: the parity of each orbit of corners
and the central cubie, from the outer layer all the way down to the
inside, are independent, and can be chosen arbitrarily. And once
they're fixed, the parity of all the other orbits is given.

By "bad" parity I assume you mean a case when the edges and corners
have different parities. Starting from the solved (even parity) 3x3x3
Cube, a slice move definitely does this; four outside edges cycle, and
the corners don't move. However, on the 3x3x3, this *is* visible,
since the face centers will also have odd parity. Moreover, the
central cube (which you can't see, of course, and isn't really there
on a real cube) also has odd parity, in a way: it has undergone an odd
number of quarter turns. On a 4x4x4, a slice move on a solved cube
changes the parity of the inside 2x2x2 corners (which you can't see)
and the edges (which you can). The parity of the outer corners is
left unchanged, since they didn't move, and the parity of the face
centers is also unchanged, since 8 of them move in two cycles of four
cubies. Then the fact that the outside edges are odd while the
outside corners are even simply means that the inside 2x2x2 corners
are also odd. That's all.

Hope that helps,

```Dr. Jim Mahoney                          mahoney@marlboro.edu
Physics & Astronomy
Marlboro College, Marlboro, VT  05344
```