The first step in finding |B| in <F,U,L,R,D> is to build a little
Start-rooted data base using only the ten generators F, F', U, U', etc.
I was able to search eight levels deep with a "quick and dirty" search.
To search deeper would take a good bit longer. I hope that eight levels
will be enough to calculate |B|. By constructing a companion
B-rooted data base, I should be able to test up to 15 levels deep.
(B is odd, so eight levels deep for each half-depth search only
gets you 15 levels total instead of 16.)
I haven't started the search for B yet, but I thought the
results so far might be of minor interest in their own right. As
I indicated before, these are actual cube positions. I haven't
figured out any way to apply conjugacy classes to this problem.
I find it a little puzzling that the branching factor is not
monotonically decreasing (c.f., level 3 to level 4).
Level Number of Local Branching Positions Max Factor0 1 0 1 10 0 10.000 2 77 0 7.700 3 584 0 7.584 4 4,434 0 7.592 5 33,664 0 7.592 6 255,320 0 7.584 7 1,933,936 7.575 8 14,635,503 7.568
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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