Date: Fri, 07 Oct 94 10:52:58 -0400 (EDT)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: Re: < U, R> Group -- Q+H
```Distance    Number     Branching       Number    Branching     Ratio of
from         of        Factor           of       Factor       Cubes to
Start   W-Conjugate                    Cubes                  W-Conjugate
Classes                                              Classes
```
``` 0          1                            1                  1
1          2          2                 6     6            3
2          5          2.5              18     3            3.6
3         14          2.8              54     3            3.857
4         41          2.929           162     3            3.951
5        122          2.976           486     3            3.984
6        365          2.992         1,457     2.998        3.992
7      1,091          2.989         4,360     2.992        3.996
8      3,256          2.984        13,016     2.985        3.998
9      9,627          2.957        38,482     2.957        3.997
10     28,282          2.938       113,094     2.939        3.9987
11     82,243          2.908       328,920     2.908        3.9994
12    235,611          2.865       942,351     2.865        3.9996
13    654,297          2.777     2,616,973     2.777        3.9997
14  1,693,858          2.589     6,774,848     2.589        3.9997
15  3,776,718          2.230    15,105,592     2.230        3.9997
16  6,058,483          1.604    24,231,019     1.604        3.9995
17  4,856,334          0.802    19,421,274     0.802        3.9992
18    961,504          0.198     3,843,568     0.198        3.997
19     11,954          0.012        47,465     0.012        3.971
20         16          0.001            54     0.002        3.375
```
```Total 18,373,824                   73,483,200                  3.9993
```

Notice that using Q+H turns instead of Q turns reduces the maximum
distance from Start from 25 down to 20.

When I first calculated God's Algorithm for <U,R> for Q turns, I calculated
it for cubes first, then for W-conjugate classes. In this case, I
really did it only for W-conjugate classes (problem is four times
smaller). The "Number of Cubes" column is then derived by calculating
the size of each W-conjugate class; no real search is needed to
obtain the number of cubes if the W-conjugate classes are already
in hand.

``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax