Date: Sun, 12 Nov 95 01:33:00 -0500
From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Magic Platonic Solids

First a correction (sorry Dave!)

# Perhaps David Badley could confirm the following orders:

The above should be "David Bagley".

I have some further comments on the "Magic Platonic Solids".

One can stretch (abuse?) the concept of the slice and anti-slice
groups of the cube to include the Megaminx (Magic Dodecahedron).
In the case of the Megaminx we can consider one-fifth turns of
opposite faces. Unfortunately my experiments with "slice" turns
on the Megaminx has not generated any spot patterns as yet.

Ben Halpern was not the only one to make a prototype of a tetrahedron
with rotating faces, as Kersten Meier made one as well.

Only 3 of the 4 generators of the Halpern-Meier Tetrahedron are
necessary to generate the 3,732,480 possible states. If we use only
2 generators we only get 19,440 possible states.

It is not possible to swap just 1 pair of corners and 1 pair of
edges, as is possible with the standard Rubik's cube.

The number of possible states of the Halpern-Meier Tetrahedron break
down like this:

```6! /2 * 2^5 * 4!/2 * 3^3 = 3,732,480
```

The number of pairs of exchanges of the 6 edges must be even.
The number of pairs of exchanges of the 4 corners must be even.
5 of the 6 edges may have any flip, the last edge is forced.
3 of the 4 corners may have any twist, the last corner is forced.

The H-M Tetrahedron is roughly comparable to the 2x2x2 cube and
the standard Skewb in terms of the number of combinations.

```Halpern's Tetrahedron    3.7*10^6       Ben Halpern, Kersten Meier
Pocket Cube (2x2x2)      3.6*10^6       Erno Rubik
Skewb                    3.1*10^6       Tony Durham
```
```-> Mark <-
```