The Prisoner of Benda

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Season 6 episode
Broadcast season 7 episode
The Prisoner of Benda
Fry-Zoidberg and Leela-Prof.png
Fry and Leela in Zoidberg's and Farnsworth's bodies.
No.98
Production number6ACV10
Written byKen Keeler
Directed byStephen Sandoval
Title captionWhat happens in Cygnus X-1 Stays in Cygnux X-1
First air date19 August, 2010
Broadcast numberS07E10
Title referenceThe Prisoner of Zenda
Additional
Commentary
(Transcript)
Transcript

Pictures

Season 6
  1. Rebirth
  2. In-A-Gadda-Da-Leela
  3. Attack of the Killer App
  4. Proposition Infinity
  5. The Duh-Vinci Code
  6. Lethal Inspection
  7. The Late Philip J. Fry
  8. That Darn Katz!
  9. A Clockwork Origin
  10. The Prisoner of Benda
  11. Lrrreconcilable Ndndifferences
  12. The Mutants Are Revolting
  13. The Futurama Holiday Spectacular
  14. The Silence of the Clamps
  15. Möbius Dick
  16. Law and Oracle
  17. Benderama
  18. The Tip of the Zoidberg
  19. Ghost in the Machines
  20. Neutopia
  21. Yo Leela Leela
  22. Fry Am the Egg Man
  23. All the Presidents' Heads
  24. Cold Warriors
  25. Overclockwise
  26. Reincarnation
← Season 5Season 7 →

"The Prisoner of Benda" is the ninety-eighth episode of Futurama, the tenth of the sixth production season and the seventh broadcast season. The Professor and Amy are cleaning up an old mind-switching machine that switches the crew members' minds, but everything goes haywire and they can't change back.

The story

A revolutionary invention allows the crew members to exchange minds, but it goes haywire and it's up to Farnsworth and the Globetrotters to fix it.

Production

According to David X. Cohen, writer Ken Keeler penned a theorem (and proof thereof) based on group theory, then used it to explain a plot twist in this episode.[1]

Writer Eric Rogers calls The Prisoner of Benda his favorite Futurama episode alongside "Jurassic Bark", "because it may be the epitome of what this series attempts to do every week: the perfect blend of science fiction and bust-a-gut humor".[2]

The theorem

Here’s an algorithm to sort out any situation.

1. First, make sure you have a buddy that you’ve never switched with, and that neither of you have switched with anybody in the group of mess-ups.

2. Make that buddy sit on the sidelines.

3. Start by switching with anyone. (In the example situation given in this episode, switch with Zoidberg into Fry’s body, for example.)

4. Switch with the one who has a mind that matches the body you’re in. Repeat until you reach the end of the closed loop. (After switching with Fry so he gets his body back, you’ll be at the end of that loop since you can’t switch with Zoidberg again.) If that wasn’t the last closed loop (i.e. is there anyone left that you’re allowed to switch with, but who aren’t in their original body?), switch with anyone of those, then repeat this step again. (In this case, there are plenty. Say you’ll switch with the professor. You’ll have Bender’s body, so switch with him. Then you’ll have the Emperor’s body, so switch with Washbucket, and so on.) When you can’t switch no more, you’ll have one closed loop left of N players, in a neat chain. (N is one from each closed loop, plus you. If you picked the same examples as above, you’d have the professor in Zoidberg’s body, you in the professor’s body and Zoidberg in your body.)

5. Bring your buddy in to switch with everyone going through that last chain in the same way you did in step four (starting with anyone). You’ll end up neatly sorted.

The total number of switches needed with this algorithm will be [number of original messed-up people] + [number of closed loops] + 2. That would be thirteen in this case since Fry and Zoidberg were in one closed loop, the others in another, and they were nine people total.

Keeler doesn’t use the sit-on-the-sidelines–idea, he brings in both Bubblegum Tate and Sweet Clyde in parallel, and he also makes it so that they don’t have to switch directly with each other—anyone have his algorithm, or did he use brute force? Here are his results. He also uses thirteen switches total.

  • Fry's body (receiving Sweet Clyde's mind) ↔ Sweet Clyde's body (receiving Dr Zoidberg's mind)
  • Dr Zoidberg's body (receiving Bubblegum Tate's mind) ↔ Bubblegum Tate's body (receiving Fry's mind)
  • Sweet Clyde's body (receiving Bubblegum Tate's mind) ↔ Dr Zoidberg's body (receiving Dr Zoidberg's mind)
  • Bubblegum Tate's body (receiving Sweet Clyde's mind) ↔ Fry's body (receiving Fry's mind)
  • Professor Farnsworth's body (receiving Bubblegum Tate's mind) ↔ Sweet Clyde's body (receiving Leela's mind)
  • Washbucket's body (receiving Sweet Clyde's mind) ↔ Bubblegum Tate's body (receiving The Emperor's mind)
  • Sweet Clyde's body (receiving Hermes's mind) ↔ Leela's body (receiving Leela's mind)
  • Bubblegum Tate's body (receiving Bender's mind) ↔ The Emperor's body (receiving The Emperor's mind)
  • Hermes's body (receiving Hermes's mind) ↔ Sweet Clyde's body (receiving Amy's mind)
  • Bender's body (receiving Bender's mind) ↔ Bubblegum Tate's body (receiving Professor Farnsworth's mind)
  • Sweet Clyde's body (receiving Washbucket's mind) ↔ Amy's body (receiving Amy's mind)
  • Bubblegum Tate's body (receiving Bubblegum Tate's mind) ↔ Professor Farnsworth's body (receiving Professor Farnsworth's mind)
  • Washbucket's body (receiving Washbucket's mind) ↔ Sweet Clyde's body (receiving Sweet Clyde's mind)

The proof

A screenshot of the proof, transcribed to the right. Click to enlarge.

First let π be some k-cycle on [n] = {1 ... n} wlog write:

π = 1  2  ...  k  k+1  ...  n
    2  3  ...  1  k+1  ...  n

Let <a,b> represent the transposition that switches the contents of a and b. By hypothesis π is generated by DISTINCT switches on [n]. Introduce two "new bodies" {x,y} and write

π* = 1  2  ...  k  k+1  ...  n  x  y
     2  3  ...  1  k+1  ...  n  x  y

For any i=1 ... k let σ be the (l-to-r) series of switches

σ = (<x,1> <x,2> ... <x,i>) (<y,i+1> <y,i+2> ... <y,k>) (<x,i+1>) (<y,i>)

Note each switch exchanges an element of [n] with one of {x,y} so they are al distinct from the switches within [n] that generated π and also from <x,y>. By routine verification

π* σ = 1  2  ...  n  x  y
       1  2  ...  n  y  x

i. e. σ reverts the k-cycle and leaves x and y switched (without performing <x,y>).

NOW let π be an ARBITRARY permutation on [n]. It consists of disjoint (nontrivial) cycles and each can be inverted as above in sequence after which x and y can be switched if necessary via <x,y>, as was desired.

Additional Info

Allusions

Trivia

Continuity

Goofs

  • Joe Gilman says there can barely fit twelve Robot Clowns in the car, but there are actually only nine.
  • Fry's and Bender's apartment look much different from earlier appearances (as in "The Late Philip J. Fry"). While previously being a small apartment (Bender's room) with a giant sideroom (Fry's), it now does not have the small apartment and it includes a kitchen. (The kitchen had been seen previously, such as in Bender's Game
  • Leela's building had been destroyed by the alien scammers in Bender's Big Score to make room for a Panda preserve.
  • It was established in the episode "Why Must I Be a Crustacean in Love" that, once a male Decapod releases his male jelly (has sex), he dies. However, in this episode, Fry, who was in Zoidberg's body, wound up having intercourse with Leela, who was in the professors body, and did not die.
    • It could be that, to die, the Decapodians need to mate with another Decapodian.

Characters

(In alphabetic order)

References

  1. ^ "In an APS News exclusive, Cohen reveals for the first time that in the 10th episode of the upcoming season, tentatively entitled "The Prisoner of Benda", a theorem based on group theory was specifically written (and proven!) by staffer/PhD mathematician Ken Keeler to explain a plot twist."
    Levine, Alaina G.. "Profiles in Versatility:". American Physics Society. Retrieved on 15 May 2010.
  2. ^ [http://www.gotfuturama.com/Information/Articles/Eric_Rogers_Interview.dhtml CGEF Interview with Eric Rogers]