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It is now known that using the qturn metric, Start has a

unique antipode in the edge group, namely Mirror-Image-

of-Edges-Flipped. The antipode is 15 qturns from Start.

Also, I have a complete data base of equivalence classes

in the edge group documenting the distance from Start for

each configuration of the edges.

It seems to me that given these two facts, some additional

distances can be determined. For example, it is possible

to determine the distance from any configuration to

Mirror-Image-of-Edges-Flipped. Let Z be a sequence of operators

that converts Start to Mirror-Image-of-Edges-Flipped, and let

A be any configuration of the edges. Then apply Z' to A, look

up the result in the data base of distances from Start, and

that will be the distance from A to Mirror-Image-of-Edges-Flipped.

The reason is quite simple. Let P be a sequence which takes

Z'(A) to Start. Then, Z'PZ takes A to Mirror-Image-of-Edges-Flipped.

This is a very nice use of conjugates.

Another consequence of this result is the following: suppose you

began with Mirror-Image-of-Edges-Flipped and performed a

breadth-first exhaustive search. Start would be antipodal, and

the number of nodes at each level of the tree would be identical

to the existing tree which begins at Start.

In addition, all of the above applies to Mirror-Image-of-Start

and Edges-Flipped with respect to each other. They are

mutually antipodal, and are 15 qturns apart. A tree built with

either at the root would have exactly the same number of nodes

at each level as the existing tree with Start at the root.

Finally, the distance of any configuration from Mirror-Image-of-Start

or Edges-Flipped can be determined. Let Y be a sequence of operators

which converts Start to Mirror-Image-of-Start, and let X be a sequence

of operators that converts Start to Edges-Flipped. Let A be any

cube. Then, the distance of A from Mirror-Image-of-Start

is the same as the distance of Y'(A) from Start, and

the distance of A from Edges-Flipped is the same

as the distance of X'(A) from Start.

I have the sensation in describing this that the Edge group is

square, with Start and Mirror-Image-of-Edges-Flipped 180 degrees

apart, and Mirror-Image-of-Start and Edges-Flipped at the other

two corners of the square.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?