Date: Sun, 21 Aug 94 08:58:30 -0400 (EDT)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ ~~~ Subject: Analysis of Turns Towards Start for Whole Cube

The following chart for level 0 through level 8 of the whole
cube using Q-turns gives the number of Q-turns for each cube which will
move towards Start. (I recently gave the same analysis for corners only.)

Columns 9 through 12 are omitted from the chart since they contain
all zeros. Any local maxima would appear in column 12.

This adds one new level known not to contain any local maxima. It would
be extremely interesting to be able to extend the chart at least to
level 12 because level 12 is known to include a local maximum.

As with the corners-only case, the chart contains almost all even numbers.
However, unlike the corners-only case, the numbers do not cluster in
the even columns. Rather, they cluster towards column 1. This means that
(close to Start, at least) most cubes have only one "good" move that
takes you closer to Start. It also serves to illustrate why "random"
moves so quickly scramble the cube.

>Number of Q-turns which Move
> Closer to Start

```Level  Total  0         1         2      3      4   5    6   7   8
Cubes
```
```0        1  1         0         0      0      0   0    0   0   0
1       12  0        12         0      0      0   0    0   0   0
2      114  0        96        18      0      0   0    0   0   0
3     1068  0       912       144     12      0   0    0   0   0
4    10011  0      8544      1368     96      3   0    0   0   0
5    93840  0     80088     12816    912     24   0    0   0   0
6   878880  0    749376    120612   8640    252   0    0   0   0
7  8221632  0   7001712   1135104  82152   2664   0    0   0   0
8 76843595  0  65391504  10645824 777936  28200  48   56   0  27
```
```Total  86049153  1  73232244  11915886 869748  31143  48   56   0  27
```
``` = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax