I learned two things about Futurama recently which added to my already deep appreciation for the show. The first is that the theme song came from a very cool song by Pierre Henry called Psyche Rock from 1967, which is on youtube. It was remixed by Fatboy Slim in an appealing way.
But what was most interesting recently was to see episode 10 of season 6, the Prisoner of Benda, a spoof of the Prisoner of Zelda but including what may be the first tv-episode publication of the proof of a relatively complex mathematical theorem in group theory as a core plot element.
The problem in the plot is that the Professor’s mind swapping machine creates an immune response which prevents swapping back in one step. So how do you get everyone back to into their original bodies? Well, as Sweet Clyde says, it takes at most two extra players [who haven’t swapped yet]. As the entire cast, including the robo-bucket, have swapped bodies, the situation is pretty complex, but fortunately one of the show’s writers, Ken Keeler, has a PhD in applied mathematics from Harvard and found a proof, which is actually shown in the show (above), and then worked in a fast montage that restores everyone.
In the following table, the heading shows the character name of the body, row 0 shows the occupant of that body by the end of the plot’s permutations and before the globetrotters start the transformations. Rows 1-7 show the steps to restore everyone to their original bodies. Each transformation was animated as a pair using the two “extra players” except the last rotation to restore Sweet Clyde and the Bucket.