Date: Tue, 22 Nov 94 13:07:05 -0500
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: <U,R> Q vs. Q+H Distances from Start

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 Start

```       0 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 Start

```            15      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