I think you're on the right track (get it?) as far as permutations. The problem I've had (and it's probably me being short-sighted) is that certain pieces (switches) not only have three endpoints instead of two, but they also can have that offshoot node either left or right-handed, depending on which side is up. So, do they take two positions in the permutation matrix?

Some of you will now be thinking of my omission: the other direction. That same switch piece can not only be flopped left-to-right, but also end-over-end. So, when we get involved with tracking male and female connectors, there's really *four* states for a switch piece, and two for more regular pieces.

I'm no mathemetician, but this seems to be the most difficult part of the problem, followed closely by differentiating acceptable collisions from unacceptable ones.