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The fact that there is only one position in <U,R> which is antipodal

for both Q moves and Q+H moves got me to thinking more generally about

the Q distance from Start as compared to the Q+H distance from Start

for any position. Here follows a cross-tabulation table giving the

number of positions in <U,R> unique up to W-conjugacy which are m moves

from Start under Q and n moves from Start under Q+H.

The table has some interesting features. The main diagonal contains

powers of 2 for distances from zero to nine. Other entries in the

upper left portion of the table (close to Start) are often small

primes times powers of 2. There are positions where allowing Q+H

moves saves nine moves (from nineteen down to ten), which is a

substantial savings in moves. There is one "discontinuity" in the

table; a position which is 25 moves from Start under Q may be 20,

19, 18, 17, or 15 moves from Start under Q+H, but not 16 moves from

Start. The longest "common distance" is nineteen; there are

positions which are nineteen moves from Start under both Q and Q+H,

but for twenty moves and above there are no positions which are

a common distance from Start.

Q+H Distance from Start0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 0 1 Q 1 1 2 1 2 D 3 2 4 i 4 1 6 8 s 5 3 16 16 t 6 1 12 40 32 a 7 4 40 96 64 n 8 1 20 120 224 128 c 9 5 80 336 512 256 e 10 1 30 280 896 1148 508 11 6 140 888 2292 2532 1004 f 12 1 42 558 2632 5688 5455 1948 r 13 5 220 1976 7433 13656 11585 3675 o 14 50 976 6475 20158 32064 24082 6387 m 15 4 297 3810 19993 52672 73401 46779 16 47 1475 13603 58642 134127 160373 S 17 3 320 6291 45381 165993 327673 t 18 39 1799 24125 141267 447893 a 19 2 268 7822 80316 403413 r 20 16 1297 26988 225778 t 21 74 4300 67328 22 1 148 8034 23 198 24 2 25

Q+H Distance from Start15 16 17 18 19 20 0 Q 1 2 D 3 i 4 s 5 t 6 a 7 n 8 c 9 e 10 11 f 12 r 13 o 14 m 15 9942 16 82146 12480 S 17 321647 116231 8729 t 18 740143 517578 98299 2066 a 19 1083499 1319438 493690 26824 42 r 20 961774 1957300 1304102 140670 566 t 21 470457 1531160 1733845 337599 2685 22 101326 545579 1018042 332435 4875 9 23 5760 58037 194963 115863 3286 3 24 23 680 4661 6032 493 3 25 1 3 15 7 1

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

If you don't have time to do it right today, what makes you think you are

going to have time to do it over again tomorrow?