Background

The “Futurama” episode “The Prisoner of Benda” begins with Professor Farnsworth finishing a body-swapping machine. The machine can swap the minds of two bodies no more than once. Amy wants to eat without making her own body fat, so she switches bodies with Professor Farnsworth. Farnsworth (in Amy’s body) believes he can switch to his original body using a third person, so he switches bodies with Bender. However, Bender (in Amy’s body) decides to commit crimes in his new body so that his original identity won’t be compromised. When Bender does performs his robbery, Farnsworth grows tired of solving the mind-swapping problem and joins a robot circus.

Meanwhile Leela switches bodies with Amy (in Farnsworth’s body) so that she can prove to Fry that Fry only loves Leela for her looks. Fry then switches bodies with John Zoidberg (an anthropomorphic crab that is also a doctor) so that he can prove to Leela (who is still in Farnsworth’s body) that Leela only loves Fry for his looks. Needless to say, the body-swapping machine creates a lot of swapped bodies, so the Harlem Globetrotters arrive at the situation to help. After some mathematical reasoning, the Globetrotters decide that everyone only requires 2 un-swapped bodies to restore everyone back to normal. After the correct body-swapping, everyone is back in their original bodies.

Is it Possible?

The question this episode poses is, “Is it possible to restore everyone back to their original bodies. If so, how many bodies are required to return everyone back to their original bodies?” The answer is: no more than two people are required. This hypothetical situation is proved by Ken Keeler, who has a PhD in applied Mathematics from Harvard. Keeler wrote this episode to introduce “Keeler’s Theorem” (or “Futurama Theorem”), which is one of the few rare instances where a show was written around a theorem. The theorem is stated as so:

Keeler's Theorem

Let A be a finite set, such that there exists elements x and y in A and x does not equal y. Then given any permutation r in A, there exists a permutation product such that, when x and y are composed with r, we can get the identity permutation [4].

There is a lot of math jargon in that theorem, but we can elaborate on this theorem. A permutation is a swap between two or more objects, and the identity permutation is the permutation where everything is in the original position. So if you consider a group of humans along a line (consider the number of humans as the finite set A). We may swap any two people, but we can swap them no more than once. In order to get everyone back in their original positions, all we would need is two people that haven’t swapped with anyone else in order to get the original line [4].

In other words, the Harlem Globetrotters were right. While the technology for swapping bodies may be later on down the road, there is a solution to getting everyone back to their original bodies. All that’s required is two people who are willing to never swap bodies with everyone else. For more information about Keeler’s Theorem, see the YouTube link below:

http://www.youtube.com/watch?v=dow7bWdr_YA

Prisoner of Benda

Bibliography

Source [12]

Background

Is it Possible?

Keeler's Theorem