Both the original note and the reply were rather lengthy, so I will
not quote the whole thing. We were at the point where we were
discussing models for cubes with edges only. Such a cube can be
modeled as a set of cosets of C, or as a subset of G or of CG. We
arrived at the following dialog which discussed the correspondence
between the two kinds of models.
On 12/10/94 at 15:12:00 Martin Schoenert said:
A similar argument applies to G[E]/C[E] except that we have to fix
an edge cubie instead of a corner cubie.
Almost. But there is a tricky problem here. Again G[E] = CG[E],
so it doesn't matter whether we talk about G[E]/C[E] (as you prefer)
or about CG[E]/C[E] (as I prefer). Again we can find a supplement
for C[E] in CG[E], namely the subgroup of all elements of CG[E]
that leave a particular edge cubie fixed. Assume that we fix the
upper-right edge cubie, then this supplement is <L[E],D[E],F[E],B[E]>.
But this does *not* respect costs. That is take an element e of CG[E].
Let r be its representative in <L[E],D[E],F[E],B[E]>, i.e., c e = r,
where c is a rotation of the entire cube. The the costs of the two
elements, viewed as elements of CG[E] is clearly the same (remember,
rotations cost nothing). But the cost of r, viewed as an element of
<L[E],D[E],F[E],B[E]> *with this generating system*, may be higher.
Indeed, *is* higher a large percentage of the time, I think.
For example take R[E] * r[E]' (where r is the rotation of the entire
cube). In CG[E] this element has cost 1. And this element lies in
<L[E],D[E],F[E],B[E]>, since it fixes the upper-right edge cubie.
But the cost of this element *with respect to the generating system
L[E],D[E],F[E],B[E]* is not 1.
We can repair this by choosing a different generating system for
<L[E],D[E],F[E],B[E]>, for example the system
So in general a model for the solution up to rotations for a
certain set <orbits>, is a supplement of C[<orbits>] in CG[<orbits>],
with a generating system that respects costs.
I guess I need to understand a little more precisely what we mean
by "respecting costs", but I have a question here. I was thinking
about this issue of representing edges only cubes by cosets vs.
representing edges only cubes as subsets of G before your note arrived,
and I thought I saw a problem.
It is well known that G[E] must have an equal number of even and
odd permutations. If we generate G[E] as <Q[E]>, it is also the case
that there are just as many positions an even distance from Start as
an odd distance from Start because the parity of the distance from
Start is the same as the parity of the permutation if we restrict
ourselves to quarter turns.
But in the computer search for God's Algorithm for edges only cubes,
there were not equal numbers of positions an even distance from Start
as an odd distance from Start. The computer search used the coset model
G[E]/C[E], where this notation means the set of cosets of C, *not* the
factor group. In and of itself, the mismatch between the number of
positions at an even distance from Start and at an odd distance from
Start should not pose a problem. It is not clear to me what it means
to speak of the "parity" of a coset of C, and half of each coset will
be even and the other half will be odd. Hence, it is not clear that
a particular coset must *a priori* be an odd or even distance from
But if we map each coset to an element of G[E], it is meaningful to
speak of the parity of the element of G[E]. And if the elements of
G[E] to which we map the cosets form a subgroup, then there must be an
equal number of odd and even elements in the subgroup
(unless they are all even?!). If the mapping respects
costs in the sense of maintaining the same distance from Start, then
I don't understand how we can avoid a conflict between the equal
number of even and odd permutations in the subgroup of G[E] and
the unequal number of even and odd distances from Start in the coset
Perhaps you could clarify your generating system and its respecting
of costs a bit further?
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU