It's a fool's errand, I suppose, but in light of our recent discussions,
I created an X-rooted data base up through level 5, where X is the
representative element of {m'Fm}. Here are the results compared to
a standard Start-rooted data base. It is most important to realize that
both data bases contain only representative elements, and that the
results with total cubes are derived from the representative elements
by calculating the sizes of the conjugacy classes.

Start-                             Repr{m'Fm}-
Rooted                             Rooted

                      Representative                   Representative
Level        Cubes      Elements               Cubes     Elements

0               1             1                 12         1
1              12             1                115         6
2             114             5              1,068        25
3           1,068            25             10,011       219
4          10,011           219             93,840     1,978
5          93,840         1,978            878,880    18,395

Performing the search in this fashion, it seems to me that there are
only four positions for which the search would look the same as for
Start -- Start itself, the Superflip, the Pons Asinorum, and the
composition of the Superflip with Pons Asinorum.

Martin Shoenert and Mark Longridge have convinced me that the Pons
Asinorum and the composition of the Superflip with Pons Asinorum
are not in the center of the cube group. But I still believe that
the search space for all four position looks essentially the same
because these are the only four positions for which the associated
symmetry group is M. That is, it is only these four positions for
which X=m'Xm for all m in M.

It was in this sense -- that the search space structure using
representative elements is the same for Start and for superflip --
that I meant that two half-depth searches using representative elements
were easy for the superflip, but would be harder for other positions.

Here is a question for Dik Winter and Mike Reid (and my apologies
if I have asked this before): have you tried your Kociemba's
algorithm programs for the composition of Pons Asinorum with
superflip? I would find the results to be *very* interesting.

Finally, as one last fool's errand, I performed the search for
the first five levels again, this time using cubes instead of
representative elements. With this last search, the results are
the same whether the root of the search is Start or something
else, which is the point both der Mouse and Martin Schoenert
were making. In this chart, "level" has to be interpreted as
"distance from root", not "distance from Start".

Start-            Repr{m'Fm}-
Rooted              Rooted

Level        Cubes                 Cubes

0               1                      1
1              12                     12
2             114                    114
3           1,068                  1,068
4          10,011                 10,011
5          93,840                 98,840

 = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax
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