Date: Sat, 04 Dec 93 23:15:30 -0500 (EST)
From: Jerry Bryan <BRYAN%WVNVM.BITNET@mitvma.mit.edu >
~~~ ~~~ 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