Date: Sun, 24 May 92 08:04:09 -0700 (PDT)
From: michael reid <reid@math.berkeley.edu >
~~~ ~~~ Subject: details ...

perhaps i should give the figures i obtained when getting my upper
bound of 39 face / 56 quarter turns in case i also have an error.
recall the method: we work in three stages:

```1. from  <U, F, R, L, B, D>  to  <U, R, F>
2. from  <U, R, F>           to  <U, R2, F2>
3. from  <U, R2, F2>         to  START
```

where we're only allowed to use moves that keep us within the
given subgroup. the number of positions that must be considered
in each stage are 253440, 15676416, and 10886400 respectively.

```stage 1:
using face turns:          using quarter turns:
distance  number           distance     number
0            1               0             1
1            9               1             6
2           90               2            39
3          852               3           276
4         7169               4          1899
5        44182               5         11245
6       131636               6         49412
7        68940               7        117221
8          561               8         70767
9          2574
stage 2:
using face turns:          using quarter turns:
distance  number           distance     number
0            1               0             1
1            2               1             2
2           12               2             8
3           72               3            36
4          420               4           158
5         2410               5           694
6        13752               6          2980
7        75796               7         12744
8       390421               8         53646
9      1735771               9        216354
10      5351383              10        799868
11      6696700              11       2477802
12      1399195              12       5310848
13        10481              13       5419046
14       1356020
15         26192
16            17
stage 3:
using face turns:          using quarter turns:
distance  number           distance     number
0            1               0             1
1            5               1             2
2           14               2             3
3           44               3             8
4          128               4            14
5          392               5            24
6         1157               6            52
7         3458               7            96
8        10057               8           176
9        29286               9           352
10        82814              10           664
11       228621              11          1248
12       591704              12          2409
13      1362497              13          4516
14      2545752              14          8519
15      3272940              15         16100
16      2260555              16         30171
17       484818              17         56140
18        12133              18        102981
19           24              19        186728
20        331234
21        563985
22        912719
23       1365051
24       1812011
25       2044832
26       1783956
27       1105488
28        450322
29         97881
30          7958
31           745
32            10
33             4
```

it would be nice if someone could confirm these figures. i have done
some testing, but not extensively.

mike