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On 04/14/95 at 16:03:31 mreid@ptc.com said:

>mark's post got me thinking ... i made a quick hack for the slice

>group (which is easy to represent by fixing the corners). my

>figures concur with his. i wanted to see the number of local maxima.

90 degree number of number of slice turns positions local maxima0 1 0 1 6 0 2 27 0 3 120 0 4 287 0 5 258 24 6 69 69as i'd hoped, there are local maxima at distance 5. one such is:

(FB') (RL') (U'D) (R2L2) = (R2L2) (F'B) (RL') (UD') = (R'L) (FB') (RL') (F'B) (U'D) = (U'D) (F'B) (RL') (U'D) (F'B) = (R'L) (UD') (F'B) (RL') (FB')(actually i think all are equivalent to this one under symmetries

of the cube.)this is especially interesting because it is a local maximum in the

full cube group (quarter turn metric) at distance 10q. according

to jerry bryan's results, there are no local maxima within 9q

of start, so this gives the closest local maximum. (there may well

be others.)

Results for the slice group under M-conjugacy:

Level Number of Number of Positions Local Maxima0 1 0 1 1 0 2 2 0 3 6 0 4 16 0 5 15 1 6 9 9

Mike's conjecture that all 24 positions which are a local maxima

at level 5 are equivalent under M-conjugation is correct.

I don't yet understand why Mike's position is a local maximum in the

full cube group. But assuming it is, it is not only the shortest

local maximum, it is the first local maximum which is not

Q-transitive (i.e, we have |{m'Xm}|=24, hence we have |Symm(X)|=2,

and the size of the symmetry groups for Q-transitive positions

must be divisible by 12.).

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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