From:

~~~ ~~~ Subject:

I said:

> Still trying to find a pattern which will

> result in 4 distinct ways, but I am not optimistic.

Jerry adds:

> As one more followup, for each symmetry group order in the above list,

> there exists at least one cube.

> That is, 96 of the 98 subgroups are symmetry groups for at

> least one cube. The two "missing" subgroups -- A and C -- are of

> order 24. But there is a third subgroup -- H -- of order 24

> (H is the set of 12 even rotations and 12 odd reflections), and there

> are cube positions whose symmetry subgroup is H. Hence, there are

> cube positions for every symmetry subgroup order.

Well, I figure Jerry is correct and so I kept looking for the magic

pattern which transforms 4 ways...

Number of different Pattern patterns ------- --------- ... 4 6 flip (UF, UR, FR, DB, DL, BL) ...

So there are 4 types of this 6 flip.

Jerry has said before:

> I believe that Dan and I have solved (sort of independently, and sort

> of working together) the problem you pose (and I give Dan the bulk

> of the credit). That is, how many cubs are there in each symmetry

> group and each symmetry class?

That sounds harder. Looks like I am specifying only the index of

the symmetry subgroup... perhaps it makes sense to find out

exactly which subgroup of M is the symmetry group of my positions.

It all sounds vaguely familar.... but it will try again tomorrow.

-> Mark <-