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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 onlyDistance Number of Number of from Start Nodes using Nodes using 1152-fold Symmetry 24-fold Symmetry0 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 1Total 851,625,008 40,874,803,200

>Results using q-turns and h-turnsDistance Number of Number of from Start Nodes using Nodes using 1152-fold Symmetry 24-fold Symmetry0 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 8,827Total 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?