   Date: Sun, 23 Oct 94 02:36:00 -0400   From: Mark Longridge <mark.longridge@canrem.com >
~~~ ~~~ Subject: Cross and X's
```-----------------------------
Possible Legal Cross Patterns
-----------------------------

Plummer Cross   (6 Cross order 3) =  8 patterns
Christman Cross (6 Cross order 2) =  6 patterns
4 Cross order 2 (sq group)        =  3 patterns
4 Cross order 4                   =  6 patterns
-----------
14 Six Cross + 9 Four Cross = 23 total legal Cross patterns
```

There are 0 total cross patterns in the swap orbit

```-------------------------
Possible Legal X Patterns
-------------------------

6 X order 3                       =  8 patterns
6 X order 6                       =  8 patterns
6 X order 2 (sq group)            =  1 pattern
4 X order 2 (sq group)            =  3 patterns
2 X order 2 (sq group)            =  3 patterns
-----------
17 Six X + 3 Four X + 3 Two X = 23 total legal X patterns
```

For a while I thought that [6 x order 3] combined with the [2 x pattern]
would make a new sort of [6 x order 6], but combining [6 x order 3] with
the [2 x pattern] is essentially the same as combining 6 x order 3 with
the pons asinorum or 6 x order 2.

```------------------------------
Possible Swap-Orbit X Patterns
------------------------------

6 X order 2                       =  6 patterns
6 X order 4                       =  6 patterns
4 X order 2                       =  6 patterns
4 X order 4                       =  6 patterns
-----------
```
```12 Six X + 12 Four X = 24 total swap-orbit X patterns
```

Some description of the swap-orbit patterns is in order.
The 6 X order 2 pattern has a 2-cycle of opposite edges and
2 sets of 2-cycles of adjacent edges.
The 6 X order 4 pattern has a 2-cycle of opposite edges and
a 4-cycle of edges of adjacent faces.
The 4 X order 2 has 2 sets of 2-cycles of adjacent edges.
The 4 X order 4 has a 4-cycle of edges of adjacent faces.

To make any of these swap-orbit patterns one would have to first
exchange any 2 edge cubies.

Interestingly, a thin line 6 X order 3 is possible on the 5x5x5
cube. No process as yet....

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