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Subject: Assorted Pyraminxi
From: mark.longridge@canrem.com (Mark Longridge)
Message-Id: <60.1049.5834.0C1D048E@canrem.com>
Date: Thu, 16 Feb 1995 00:11:00 -0500
Organization: CRS Online  (Toronto, Ontario)

MS>
I am fairly certain that the Pyraminx is a regular tetrahedron.  In the
solved state each of the four faces shows only one of the four colours.

ML>
This is correct for all of the *tetrahedral* pyraminxes with only
one small exception: the Star Pyraminx has all middle pieces the
same colour. Meffert used the word "pyraminx" as a prefix to just
about all the puzzles he either conceived or planned to market.

MS>
The Pyraminx Star was described as a Pyraminx without the centers.
So I guess each face of the Pyraminx Star looks as follows.

                   +
                  / \
                 /   \
                /  V  \
               /       \
              +---------+
             / \       / \
            /   \  M  /   \
           /  E  \   /  E  \
          /       \ /       \
         +---------*---------+
        / \       / \       / \
       /   \  M  /   \  M  /   \
      /  V  \   /  E  \   /  V  \
     /       \ /       \ /       \
    +---------+---------+---------+

ML>
Actually the above diagram is a good representation of a head-on
view of the popular or standard pyraminx (I've taken the liberty
of embellishing it a little).

There are 4 Vertices (3 colours), 6 Edges (2 colours) and 12 Middle
pieces (single colur) so there are 12 + 12 + 12 = 36 facelets.

The tips (or small vertices) can rotate independently, and the larger
turn includes the rotaion of the adjacent 2 edge pieces and single
middle piece. The small tips each have 3 positions, it's adjacent
middle piece also has 3 positions, and the 6 edges obey the same
basic laws as the cube, so there are:

3^4 * 3^4 * (6!/2) * (2^6 /2) = 75,582,720 combinations or
approximately 75.5 million (993,120 for the snub version)

The math for the pyraminx octahedron is very similar, though
it has 4 positions for the 6 vertices and middle pieces
and 12 edges:

4^6 * 4^6 * (12!/2) * (2^12/2) = 8,229,184,826,926,694,400 or
approximately 8.2 quintillion.

So the snub pyraminx (or if you prefer "The Pyraminx Snub") would
look like:

              +---------+
             / \       / \
            /   \     /   \
           /     \   /     \
          /       \ /       \
         +---------*---------+
          \       / \       /
           \     /   \     /
            \   /     \   /
             \ /       \ /
              +---------+

One could imagine snub octahedrons as well.

MS>
I have no idea what Pyraminx Senior and the Pyraminx Master look like.

ML>
They are visually indistinguishable from the standard pyraminx,
however information on the Senior Pyraminx is exceedingly sketchy.

I've never seen a photograph of a Master Pyraminx in the middle
of an edge turn so I rather doubt a working prototype was ever
made, but you never know...

I'll take a stab at one more calculation...

The pyraminx hexagon has 12 corners and 18 edges and 8 centres.
Each side has 13 facelets so there are 13 * 8 = 104 total facelets.
12!/2 * 3^11 * 16! * 2^15 = 29,087,761,395,446,975,811,708,518,400,000
or approximately 29 nonillion (29^30).

-> Mark <-