From:

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Here are my God's Algorithm results for the corners of the

3x3x3 cube. I explained in the last post what I mean by

1152-fold symmetry and 24-fold symmetry. The 1152-fold

symmetry is what I actually calculated. In this particular

case, I did not do the 24-fold symmetry calculations myself

based on the size of the equivalence classes as I did

with the 2x2x2 cube. Rather, I went back and found the

figures in the Cube-Lover archives (Dik Winter's post).

Results Using Both q-turns and h-turnsDistance Number of Number of from Nodes using Nodes using Start 1152-fold 24-fold symmetry symmetry0 1 1 1 2 18 2 9 243 3 71 2,874 4 637 28,000 5 4,449 205,416 6 24,629 1,168,516 7 113,049 5,402,628 8 433,611 20,776,176 9 947,208 45,391,616 10 316,823 15,139,616 11 1,481 64,736

>Results Using Only q-turnsDistance Number of Number of from Nodes using Nodes using Start 1152-fold 24-fold symmetry symmetry0 1 1 1 1 12 2 5 114 3 24 924 4 149 6,539 5 850 39,528 6 4,257 199,926 7 16,937 806,136 8 57,800 2,761,740 9 180,639 8,656,152 10 466,052 22,334,112 11 676,790 32,420,448 12 392,558 18,780,864 13 45,744 2,166,720 14 163 6,624Results Using Only h-turnsDistance Number of from Nodes using Start 1152-fold symmetry0 1 1 1 2 2 3 4 4 3

It turns out that the maximum distance from Start is the same

for the corners of the 3x3x3 cube as it is for the 2x2x2 cube.

I found this rather surprising, although the archives of

Cube-Lovers do provide a reasonable explanation. I am just

going to have to go back and read it five or ten times until

I fully understand it. In any case, I was curious about the

following question. Suppose you are N moves from Start on the

corners of the 3x3x3. How many moves from Start would you be

on the 2x2x2 if the 2x2x2 was in the same configuration as the

corners of the 3x3x3 where you currently were. As it turns out,

I stored the results for the 2x2x2 in the same data base as

I stored the results for the corners of the 3x3x3, so the

question was easy to answer. Here are the results.

Corresponding Distances from Start

Using Both q-turns and h-turns2x2x2 Corner of 3x3x3 Number Distance from Distance from of Nodes Start Start0 0 1 2 2 4 2 1 1 2 2 2 3 4 4 3 2 2 5 3 12 4 18 5 3 3 3 55 4 106 5 41 4 4 508 5 457 6 38 5 5 3,948 6 1,237 7 2 6 6 23,354 7 1,992 8 20 7 7 111,055 8 3,242 9 20 8 8 430,349 9 5,460 10 62 9 9 941,728 10 3,770 11 2010 10 312,991 11 45 11 11 1,416

>Corresponding Distances from Start

> Using Only q-turns2x2x2 Corner 3x3x3 Number Distance from Distance from of Nodes Start Start0 0 1 2 1 4 2 6 1 1 1 1 3 2 5 2 22 4 4 10 6 63 3 22 5 46 7 4 4 4 137 6 145 5 5 802 7 356 6 6 4,105 8 474 77 16,577 9 838 8 57,326 10 24 12 24 9 9 180,556 11 14810 10 466,028 12 192 11 11 676,642 13 144 12 12 392,342 13 13 45,600 14 14 163

>Corresponding Distances from Start

> Using Only h-turns2x2x2 Corner of 3x3x3 Number Distance from Distance from of Nodes Start Start0 0 1 2 1 1 1 1 3 1 2 2 1 3 3 3 4 4 3

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?