While I have not solved this problem, I remember a friend who worked on it back in the 60's/70's (I don't recall which). He had worked out a program for the IBM 1130 (in the language of the day: Fortran!) that went thru and found combinations of the cubes that would NOT work. It seems as though if you put the faces on correctly you can make it an insane puzzle. Now, I don't know the result of this musing, but given a set of cubes, and their available colors, making an "impossible" set might be a more difficult task. As I remember it they pryed apart the cubes and snapped them back to give the set to the math teacher or something like that. Good luck!