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Date: Sat, 4 Dec 1993 23:15:30 EST
From: "Jerry Bryan"
To: "Cube Lovers List"
Subject: God's Algorithm for the Corners of the 3x3x3
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-turns
Distance Number of Number of
from Nodes using Nodes using
Start 1152-fold 24-fold
symmetry symmetry
0 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-turns
Distance Number of Number of
from Nodes using Nodes using
Start 1152-fold 24-fold
symmetry symmetry
0 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 6,624
Results Using Only h-turns
Distance Number of
from Nodes using
Start 1152-fold
symmetry
0 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-turns
2x2x2 Corner of 3x3x3 Number
Distance from Distance from of Nodes
Start Start
0
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 20
10
10 312,991
11 45
11
11 1,416
Corresponding Distances from Start
Using Only q-turns
2x2x2 Corner 3x3x3 Number
Distance from Distance from of Nodes
Start Start
0
0 1
2 1
4 2
6 1
1
1 1
3 2
5 2
2
2 4
4 10
6 6
3
3 22
5 46
7 4
4
4 137
6 145
5
5 802
7 356
6
6 4,105
8 474
7
7 16,577
9 83
8
8 57,326
10 24
12 24
9
9 180,556
11 148
10
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-turns
2x2x2 Corner of 3x3x3 Number
Distance from Distance from of Nodes
Start Start
0
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?