From:

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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 RootedRepresentative 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 RootedLevel 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 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU