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 StartLevel Total 0 1 2 3 4 5 6 7 8 Cubes0 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 27Total 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 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?