Date: Sun, 05 Dec 93 00:01:08 -0500 (EST)
From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
~~~ ~~~ Subject: God's Algorithm for the Edges of the 3x3x3

Here are my God's Algorithm results for the edges of the
3x3x3 cube. I explained in the last post what I mean by
1152-fold symmetry. All results below are for 1152-fold
symmetry. I am working on the 24-fold case, but I am not
quite done. The 24-fold case is just a matter of determining
the sizes of the equivalence classes in the 1152-fold case.

One item of terminology needs to be explained. Several
people, including myself, have posted results for the
2x2x2 cube and for the corners of the 3x3x3 cube. If you
take the term "corners of the 3x3x3 cube" absolutely
literally, it is completely isomorphic to the 2x2x2 cube.
When people have posted results for the "corners of the
3x3x3 cube", they all (including myself) really mean
"corners plus centers of the 3x3x3". See below:

```--------------      ---------------------    -------------------
|  x  |  x   |      |  x   |      |  x  |    |  x  |     |  x  |
|     |      |      |      |      |     |    |     |     |     |
|-----|-------      -------|------|------    ------|-----|------
|  x  |  x   |      |      |      |     |    |     |  x  |     |
|     |      |      |      |      |     |    |     |     |     |
--------------      -------|------|------    ------|-----|------
2x2x2           |  x   |      |  x  |    |  x  |     |  x  |
|      |      |     |    |     |     |     |
---------------------    -------------------
Corners of 3x3x3       Corners + Centers
```

Thus, when I say I have solved the "edges of the 3x3x3", I need
to clarify what I mean. I have solved the "edges without the
centers". I suppose my next project will be "edges with the
centers". Unfortunately, "edges with the centers" is a twenty-four
times larger problem than is "edges without the centers". "Edges
without the centers" took about a year running 24 hours a day,
7 days a week, on two machines. I am going to have to rethink
"edges with the centers" before I start. I don't want it to take
24 years.

```---------------------    -------------------
|      |  x   |     |    |     |  x  |     |
|      |      |     |    |     |     |     |
-------|------|------    ------|-----|------
|  x   |      |  x  |    |  x  |  x  |  x  |
|      |      |     |    |     |     |     |
-------|------|------    ------|-----|------
|      |  x   |     |    |     |  x  |     |
|      |      |     |    |     |     |     |
---------------------    -------------------
Edges without Centers     Edges with Centers
```

>Results using q-turns only

``` Distance                 Number of
from Start                Nodes using
1152-fold Symmetry
```
``` 0                             1
1                             1
2                             5
3                            25
4                           215
5                         1,860
6                        16,481
7                       144,334
8                     1,242,992
9                    10,324,847
10                    76,993,295
11                   371,975,385
12                   382,690,120
13                     8,235,392
14                            54
15                             1
```

>Results using q-turns and h-turns

``` Distance                 Number of
from Start                Nodes using
1152-fold Symmetry
```
``` 0                             1
1                             2
2                             9
3                            75
4                           919
5                        11,344
6                       139,325
7                     1,664,347
8                    18,524,022
9                   167,864,679
10                   582,489,607
11                    80,930,364
12                           314
```
``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax