In Jerry's message from
Date:      Mon, 16 Oct 1995 23:08:02 -0400 (EDT)
Subject: The Correctness of Large Seaches

In light of the above discussion, I thought it might be appropriate
to summarize some background about my larger searches. I want to
indicate which ones of them seem pretty well verified, and which ones
of them might benefit from further study.

<U,R>           -  I believe this one is ok.  I ran the q turn case
                   with and without conjugacy classes.  Upon
                   expanding the conjugacy classes, the results
                   matched the results without conjugacy classes.
                   Also, the results matched results posted by
                   Mark Longridge as far as they went (although
                   my anomaly with corners-only suggests that
                   such matching doesn't prove very much).

Well, I did get up to 12 q turns deep ;-)
Good enough for 2 half deep searches...

But there is another possible verification method by counting the
number of even positions and odd positions and totalling them.

Analysis of < U, R > group on 3x3x3 cube by Parity
--------------------------------------------------

Even Positions          Odd Positions
--------------          -------------

 0            1          1           4
 2           10          3          24
 4           58          5         140
 6          338          7         816
 8        1,970          9       4,756
10       11,448         11      27,448
12       65,260         13     154,192
14      360,692         15     827,540
16    1,851,345         17   3,968,840
18    7,891,990         19  13,659,821
20   18,471,682         21  16,586,822
22    8,039,455         23   1,511,110
24       47,351         25          87
     ----------             ----------
     36,741,600             36,741,600

This is almost Cube Philosophy... how can we be certain about the true
nature of God's Algorithm? How can we be certain our cube databases
are completely accurate? I suppose it is not really a big problem
as long as the various cube programs all agree, and a human
observer executes such processes on a real cube.

-> Mark <-