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I suppose it's time for a few observations on symmetry. After all,

tomorrow is the thirteenth anniversary of "Symmetry and Local Maxima."

As Jerry Bryan notes, we can perform the "R" turn by rotating the cube

to put the R face in front, performing "F", and undoing the rotation.

But we can also perform "R'" by reflecting the cube in a left-to-right

mirror, performing "L", and undoing the reflection. Thus conjugation

can be extended to use the 48-element group of rotations and

reflections, which we call M.

In the absence of face centers, there is another kind of reduction

that takes account of the 24 possible positions of the resulting

collection of edges in space. So two positions X and Y are considered

equivalent if

X = m' Y m c

where m is a rotation or reflection in M, and c is a rotation.

My understanding of Jerry Bryan's method is that he combines "m c"

into a single rotation or reflection, and factors out the reflection

on both sides. It seems to me that what he calls a a "color rotation"

is premultiplication, while a "cube rotation" is postmultiplication.

[I am somewhat uncertain of this, because it doesn't explain how there

can be a 1252-element symmetry group when face centers are present, so

perhaps I'm missing something crucial.]

But I think we are at least conceptually better off dealing with M

when dealing with conjugation, because it takes account of the

essential similarity between clockwise and anticlockwise turns. Alan

Bawden mentioned back in 1980 that certain positions with sufficient

symmetry were local maxima (in terms of distance from start), on the

grounds that any clockwise or anticlockwise turn gives us essentially

the same position. Jim Saxe and I formalized the notion in a paper

entitled "Symmetry and Local Maxima" that we posted on 14 December

1980. [You can find it in /pub/cube-lovers/cube-mail-1.Z on

FTP.AI.MIT.Edu].

We had some hope that some of these local maxima might turn out to be

global maxima. My hopes for that have been somewhat low in recent

years. That is perhaps my best excuse for not noticing immediately

that the single global maximum for the edge group turns out to be one

of these symmetric local maxima. In fact, all four of the positions

with 24-element equivalence classes appear in the list of M-symmetric

positions.

The paper on Symmetry and Local Maxima also catalogues the positions

that have 48-element equivalence classes and 72-element equivalence

classes. The The former are the H-symmetric positions, "Six-H" and

"Six-H with all edges flipped". The latter are the twelve T-symmetric

positions. For T-symmetry, the set of flipped edges may be any of

{none, girdle-edges, off-girdle-edges, or all}; the set of edges

exchanged with their antipodes may be any of the four as well. But if

we choose "none" or "all" for all both choices we get one of the four

M-symmetric positions with 24-element equivalence classes, so only

twelve of the sixteen possibilities have 72-element equivalence

classes.

With regard to the edge cube, I should mention that no one has

mentioned a 9 QT process for the all-flip nor a 15 QT process for the

pons-asinorum-all-flip. Of course, the latter would be somewhat more

interesting, being the longest optimal sequence.

Dan Hoey

Hoey@AIC.NRL.Navy.Mil