[next] [prev] [up] Date: Tue, 09 Aug 94 01:48:00 -0400
[next] [prev] [up] From: Mark Longridge <mark.longridge@canrem.com >
[next] ~~~ [up] Subject: < U, R> Group

Well I decided to pull a "Jerry Byran" and take another look at
some cube results, plus take a look at some new groups.

Analysis of the 3x3x3 squares group
-----------------------------------
                  (h only)      branching
Moves Deep       arrangements    factor      loc max (h only)
 0                    1           --             0
 1                    6           6              0
 2                   27           4.5            0
 3                  120           4.444          0
 4                  519           4.325          0
 5                1,932           3.722          0
 6                6,484           3.356          1  (6 X pattern)
 7               20,310           3.132          0
 8               55,034           2.709         65
 9              113,892           2.069      1,482
10              178,495           1.567      7,379
11              179,196           1.004     25,980
12               89,728           0.501     50,320
13               16,176           0.180     11,328
14                1,488           0.092        912
15                  144           0.096        144
                -------                     ------
                663,552                     97,611
Analysis of the 3x3x3 <U, R> group
----------------------------------
                                          branching
Moves Deep       arrangements (q only)     factor
 0                    1                      --
 1                    4                       4
 2                   10                       2.5
 3                   24                       2.4
 4                   58                       2.416
 5                  140                       2.413
 6                  338                       2.414
 7                  816                       2.414
 8                1,970                       2.414
program starts to really bog down after this...

I leave it to Jerry or Dan to check my results. I checked up to 2
moves deep by hand and verified 10 different positions. What I don't
understand is how Jerry manages to look at so many cube positions:

On full 3x3x3 cube,   7     100,803,036     13.231  (new)

Using 10 bytes to store a single cube position would still
need over 1 billion bytes, or am I missing something?

I also used GAP (quite a good program) to calculate the size of
< U1, R1 > on the magic dodecahedron: 7,999,675,084,800.

Once again, I welcome any verification.

-> Mark <-

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