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According to Singmaster, there are 12 disjoint orbits (his term) of cube

positions. The article I read doesn't give a derivation, but the basic

idea is clear. It should be possible to characterize the orbit of a given

position as a function of the transformations that would have had to

be done on each cubie ( as if it could be moved alone ) to bring it

to its proposed position. Given such a charaterization, one could

declare positions fed to :CUBE illegal without trying to solve the

problem.

This relates to my thoughts on the problem of inversion of an unknown

transformation. Suppose you are given a Cube that is N unit moves

from home, determine the correct sequence to get it there. I think

a representation of the Cube in terms of combined tranformations on

the cubies is the place to start.

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