Date: Sun, 05 Dec 93 20:54:04 -0500 (EST)
From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
~~~ Subject: God's Algorithm, 24-fold Symmetry, Edges of 3x3x3

I just finished tabulating the results with 24-fold symmetry for
the edges of the 3x3x3 cube. I have added them to the table
I posted earlier today which had 1152-fold symmetry.

A couple of reminders. In the case of 1152-fold symmetry, most but not
all of the equivalence classes have exactly 1152 elements. In the case
of 24-fold symmetry, every equivalence class has exactly 24 elements.
Thus, you can almost, but not quite, get from the 1152-fold column
to the 24-fold column by multiplying by 48. Also, my program actually
generated the 1152-fold column. However, it did not generate the
24-fold column. That would have taken far too long. Rather, I generated
the 24-fold column from the 1152-fold column by determining the sizes
of all the equivalence classes. Finally, note that the total figure
for the 24-fold symmetry column can be calculated as
40,874,803,200 = [12!(2^12)/2] / 24, so the total is correct.

Results using q-turns only

``` Distance            Number of            Number of
from Start           Nodes using          Nodes using
1152-fold Symmetry    24-fold Symmetry
```
``` 0                        1                   1
1                        1                  12
2                        5                 114
3                       25               1,068
4                      215               9,759
5                    1,860              88,144
6                   16,481             786,500
7                  144,334           6,916,192
8                1,242,992          59,623,239
9               10,324,847         495,496,593
10               76,993,295       3,695,351,994
11              371,975,385      17,853,871,137
12              382,690,120      18,367,613,703
13                8,235,392         395,043,663
14                       54               1,080
15                        1                   1
```
```Total             851,625,008      40,874,803,200
```

>Results using q-turns and h-turns

``` Distance            Number of          Number of
from Start           Nodes using        Nodes using
1152-fold Symmetry   24-fold Symmetry
```
``` 0                        1                 1
1                        2                18
2                        9               243
3                       75             3,240
4                      919            42,359
5                   11,344           538,034
6                  139,325         6,666,501
7                1,664,347        79,820,832
8               18,524,022       888,915,100
9              167,864,679     8,056,929,021
10              582,489,607    27,958,086,888
11               80,930,364     3,883,792,136
12                      314             8,827
```
```Total              851,625,008    40,874,803,200
```
``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax
837 Chestnut Ridge Road                     BRYAN@WVNVM
Morgantown, WV 26505                        BRYAN@WVNVM.WVNET.EDU
```

If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?