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 maxima

0              1              0
1              6              0
2             27              0
3            120              0
4            287              0
5            258             24
6             69             69

as 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 Maxima

0              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
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