Subgroup Sizes of the Pyraminx Octahedron
------------------------------------------

8 * 9 = 72 facelets (triangles)

The standard Pyraminx Octahedron has 8 faces, 6 vertices, and 12 edges.
It's vertices rotate. One may imagine a "Master" Pyraminx Octahedron
with edge AND face rotations as well.

Christoph Bandelow has a version of the Pyraminx Octahedron (I call
it "Octa" for short) which has no tips.

Size of Groups without rotating vertex tips:

Name            Subgroup                         # of Elements
----            --------                         -------------

OCT1            <U>                                          4
OCT2            <U, D>                                      16
OCT3            <U, D, F>                          116,121,600
OCT4            <U, D, F, B>                   613,312,204,800
OCT5            <U, D, F, B, L>            502,269,581,721,600
OCT6            <U, D, F, B, L, R>       2,009,078,326,886,400

Size of Groups with rotating vertex tips:

Name            Subgroup                         # of Elements
----            --------                         -------------

OCT1            <U>                                         16
OCT2            <U, D>                                     256
OCT3            <U, D, F>                        7,431,782,400
OCT4            <U, D, F, B>               157,007,924,428,800
OCT5            <U, D, F, B, L>        514,324,051,682,918,400
OCT6            <U, D, F, B, L, R>   8,229,184,826,926,694,400

Approximately  8.2 * 10^18  ..so still less than the 3x3x3 cube

The number of elements increases by a factor of 4^N for
each successive group if we include the trivial vertex rotations.

A Skewb Summary
---------------

Without repeating Martin's results on the skewb, (which I concur
with) here is a quick summary on Skewb facts:

It is impossible for any face piece to turn in place 90 degrees.
It is impossible to flip a single face piece 180 degrees.
It is impossible to transpose 2 face pieces.

The Skewb has no non-trivial centre.
The SuperSkewb has non-trivial centre with all 6 face pieces
rotated 180 degrees.

The Mystery of the Five Pyraminxi
---------------------------------

Or perhaps that should be Pyraminxes... but I can not resist
comparing the Five Pyraminxes to the Five Wizards of J.R.R Tolkien,
due to their mysterious nature.

We are probably all familar with the Popular Pyraminx created
by Uwe Meffert. What really confounds me is that Dr. Ronald Turner-
Smith kepts referring to the 5 pyraminxes in ad literature and
his book "The Amazing Pyraminx". The Master Pyraminx I understand,
it has all the basic properties of the standard popular pyraminx
plus all 6 of it's edges can rotate 180 degrees (which flips one
edge, transposes 2 tips, and swaps 2 pairs of interior edge pieces)
giving a total number of permutations of 446,965,972,992,000.

Then there is the mysterious "Senior Pyraminx" (this is like
Tolkien's Blue Wizards no one knows about). I can only speculate
on the properties of the Senior Pyraminx having never read a
description, and never seen the physical puzzle itself. The only
fact on the Senior Pyraminx I am sure about is that it has less
permutations than the Master Pyraminx. My theory is that the
Senior Pyraminx has all the properties of the standard pyraminx
plus it can rotate SOME of it's edges but not all 6 like the
Master Pyraminx (perhaps one or two?).

Perhaps Mr. Singmaster, who has seen magic solid variants from
all over the world, can shed some light on the matter!

-> Mark <-
Email: mark.longridge@canrem.com