From:

Subject:

i've also run an exhaustive search on the subgroup

H = <U, D, F2, R2, B2, L2>. here are the number of

positions at each distance.

distance quarter turns face turns

0 1 1 1 4 10 2 10 67 3 36 456 4 123 3079 5 368 19948 6 1192 123074 7 3792 736850 8 11263 4185118 9 34352 22630733 10 102638 116767872 11 287320 552538680 12 810144 2176344160 13 2261028 5627785188 14 5941838 7172925794 15 16291708 3608731814 16 41973415 224058996 17 107458884 1575608 18 269542476 1352 19 628442876 20 1367654200 21 2613422312 22 3997726648 23 4444701268 24 3661653732 25 1906936668 26 407132392 27 34358944 28 1664168 29 14840 30 160

a position at distance 18 face turns was exhibited by hans kloosterman

on may 30 1992. (he also found three others that differ only in the

middle layer edges.) it was then observed by dik winter (also on may 30

1992) that kociemba's algorithm took exceptionally long for this position.

however, this does not appear to be the case for most of the antipodes.

(i will give the antipodes for each metric in separate messages.)

the 4 positions found by kloosterman are also antipodes in the quarter

turn metric, and, up to symmetry, are the only positions which are

antipodal in both metrics. hmmm...

elements of H are described by triples (c, e, m), where

c = corner permutation,

e = U D edge permutation,

m = middle layer edge permutation,

and the total parity is even. there are

8! = 40320 corner configurations, 8! = 40320 U D edge configurations and 4! = 24 middle layer edge configurations,

for a total of 40320 * 40320 * 24 / 2 = 19508428800 positions.

if we divide by symmetry along the corner coordinate, we get 2768 corner

configurations (of course we get the same number if we divide by symmetry

along the U D edge coordinate), so we can reduce to 1339269120 positions.

at 2 bits per configuration, this requires 327 megabytes, which is too

large.

however, if we also divide out by inversion, we can reduce the number of

corner configurations to 1672, the total number of positions to 808980480,

and the memory required to 200 megabytes. this is still a lot, but is

within reach.

the calculations were done on the same machine: DEC 3000 alpha 700,

configured with 256 Mb RAM. run times were much more modest:

10 hours for quarter turns, 7.5 hours for face turns.

mike