The Prisoner of Benda

"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.

Act I:"We're just the people this Mind-Switcher was made for by us!"
Bender is watching the news, and finds out that Emperor Nikolai of the Robo-Hungarian Empire is visiting New New York in his yacht. He plans on stealing the Emperor's crown, but he cannot get his co-workers to help him by doing specific jobs (such as Amy seducing the guards). Meanwhile, Amy and the Professor put the finishing touches on the Mind-Switcher. The Professor reveals that he had always wanted to live life to the extreme while he was young, and Amy reveals that she still has that demonic food lust from way back, despite having obtained an elegant figure since 2997. They decide to switch minds, so that the Professor could use Amy's young body to do extreme sports, and Amy could eat all she wants in the Professor's old body, since he was so thin. However, when they try to switch back into their normal bodies, they find out that the Mind-Switcher only works once with the same pair of people. They try to solve this problem by putting Farnsworth's mind in Bender's body, and Bender's mind in Amy's body.

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.

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".

The inversion
Keeler uses the method outlined in the proof to restore all minds to their respective bodies. Here are his transpositions. He uses thirteen switches total, which is the least possible number of switches in this scenario.


 * 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


First let π be some k-cycle on [n] = {1 ... n} WLOG [without loss of generality] write: π = 1 2  ...  k  k+1  ... n    2  3  ...  1  k+1  ... n

Let  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 = (  ... ) (  ... ) () ()

Note each switch exchanges an element of [n] with one of {x,y} so they are all distinct from the switches within [n] that generated π and also from . 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 ).

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 , as was desired.

Reception
In its original American broadcast, "The Prisoner of Benda" was viewed by an estimated 1.774 million households a down of 150,000 viewers since A Clockwork Origin

Allusions

 * The title of the episode as well as Bender's plot of posing as an emperor is a reference to the 1894 novel (as well as its numerous adaptions).
 * The title caption refers to, which is a black hole. If nothing can escape from a black hole, then clearly anything that happens in Cygnus X-1 will stay in Cygnus X-1.
 * Leela makes several mentions of movies, possibly the  series.
 * Bender in Amy's body convinces Emperor Nikolai that he is a robot by dancing 's famous.
 * Hermes says that Amy (in the body of Leela) "makes look like regular Albert".
 * Big Bertha, a robot with a cannon stomach, is a reference to the German.
 * The Idea of one-way-mind-swapping-machine are very similar to the one in episode 2x17 of.
 * The concept of a member of royalty temporarily switching lives with an ordinary person is similar to the story.
 * The UN Conference is portrayed as nightclub act in Vegas with the opening act delivered by "Peaches and Herzegovina" which is a play on both the name of Eastern European nation and American disco/soul duo ""

Trivia

 * This is the third episode of season 6 to use a cold opening (alongside "Rebirth" and "That Darn Katz!"). Cold openings were previously used most prominently in seasons 1 and 2.
 * This is the third animated TV show in which a character voiced by John DiMaggio switches bodies with someone and the voice moves with the character into the new body. The other two were the "Kim Possible" episode "Mind Games," which originally aired in 2002, with DiMaggio voicing Dr. Drakken and the "Penguins Of Madagascar" episode "Roger Doger", aired in 2009, with DiMaggio voicing Rico.
 * When Amy in the Professor's body is eating ribs, she says "I have died" in Cantonese

Continuity

 * Yet again Amy's childhood obesity is a theme, previously mentioned in episodes such as "Teenage Mutant Leela's Hurdles" and "Into the Wild Green Yonder".
 * Bender claims he is sixty percent storage space. A recurring joke is that someone claims Bender is a certain percent of something, with the total by now adding up to much more than 100 %.
 * Fry seems to be excited by the idea of Leela's mind in Amy's body, a revisiting to the Fry-Amy relationship.
 * The Professor calls Bender's body invincible, though only a few episodes earlier ("Lethal Inspection") we found out that it wasn't.
 * The episode brings a big step in the Fry-Leela relationship, as they have sex for the first time known, though it should be noted that neither is in their own body at the time.
 * This is the first time the two have had intercourse in canon, but in "Anthology of Interest I", during the simulation "Dial L For Leela", Leela was forced to keep Fry silent about the recent murders she committed by having sex with Fry, while in their own bodies. However, "Anthology Of Interest 1" can't be regarded as canon.
 * The Professor had invented a mind-switching device back in Bender's Game, but is seen here as only having just completed it. However, it may have been operational, just not perfected at that time.

Goofs

 * Joe Gilman says there can barely fit twelve Robot Clowns in the car, but there are actually only nine.
 * That's per cubic meter. If the clown car was less than a cubic meter in volume you could achieve 12 Robot Clowns per meter with only 9.
 * 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 could have been rebuilt.
 * 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.
 * It could be that, Leela was keen to use her new appendage, therefore it would be Farnsworth male jelly, not Decapod, that was released.
 * Mating and having sex are not necessarily the same thing for all species. The releasing of male jelly might be voluntary.
 * After everyone swaps back into their original bodies Hermes is as fat as usual.

Characters

 * 21st Century Girl
 * Amy
 * Debut: Bassor
 * Debut: Big Bertha
 * Debut: Chainsaw Robot
 * Bender
 * 'Sweet' Clyde Dixon
 * Ethan 'Bubblegum' Tate
 * Fry
 * Hermes
 * "Fishy" Joseph Gilman
 * Debut: Japanese UN Diplomat
 * Leela
 * Linda
 * Morbo
 * Debut: Emperor Nikolai
 * Debut: Flavia
 * Professor Farnsworth
 * Richard Nixon's head
 * Debut: Robot Acrobat
 * Debut: Robot Clowns
 * Scruffy
 * Debut: TV Robot
 * Debut: Wash Bucket
 * Zoidberg