I've seen the "Nuts" puzzle; I think it is indeed "Drive you nuts",
but I'm not sure. I seem to recall that not every (any?) piece
had all six numbers on it; that is, I think one or more numbers
were duplicated on one or more pieces. This of course opens up
the search space even more in trying to find a set of pieces with
exactly one solution. Also, rather than consider which sets
of pieces (or what fraction of possible sets of pieces) have unique
solutions, I think it makes more sense to choose a solution and see
how likely it is that the pieces forming that solution have another
solved orientation (other than trivial rotations of the whole puzzle).
That way you at least eliminate all the sets that have NO solution.