[next] [prev] [up] Date: Sun, 05 Dec 93 20:54:04 -0500 (EST)
[next] [prev] [up] From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
[next] ~~~ [up] 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?


[next] [prev] [up] [top] [help]