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{{Use American English|date=January 2025}}
{{Use mdy dates|date=January 2025}}

{{Infobox Futurama episode
| image           = 
| caption         = 
| season          = 6 <!-- Please do not change. Production order, not airing order, is used here. -->
| episode         = 10 <!-- Please do not change. Production order, not airing order, is used here. -->
| director        = Stephen Sandoval
| writer          = [[Ken Keeler]]
| production      = 6ACV10
| airdate         = {{Start date|2010|08|19}}
| guests          = 
| opening_caption = What happens in [[Cygnus X-1]] stays in Cygnus X-1
| opening_cartoon = 
| prev            = [[A Clockwork Origin]]
| next            = [[Lrrreconcilable Ndndifferences]]
}}

"'''The Prisoner of Benda'''" is the tenth episode in the <!-- Please do not change. Production order, not airing order, is used here. --> [[Futurama season 6|sixth]] [[List of Futurama episodes|season]] of the American animated television series ''[[Futurama]]'', and the 98th episode of the series overall. It aired on [[Comedy Central]] on August 19, 2010.  In the episode, [[Hubert J. Farnsworth|Professor Farnsworth]] and [[Amy Wong|Amy]] build a machine that allows them to switch minds so that they may each pursue their lifelong dreams. However, they learn that the machine cannot be used twice on the same pairing of bodies. To try to return to their rightful bodies, they involve the rest of the crew in the mind switches, leaving each member free to pursue their own personal endeavors in a different crew member's body. The episode is composed of multiple subplots, with the main subplot being [[Bender (Futurama)|Bender]] attempting to steal a crown, but ending up switching places with the Robo-Hungarian emperor.

The episode was written by [[Ken Keeler]] and directed by Stephen Sandoval and was met with acclaim from critics. The issue of how each crew member can be restored to their correct body given the limitation of the switching device is solved in the episode by what [[David X. Cohen]] described in an interview as a mathematical [[theorem]] proved by Keeler, who has a [[Ph.D#United States|Ph.D]] in Mathematics. The title and the story's main subplot is a reference to the 1894 [[adventure novel]] ''[[The Prisoner of Zenda]]'' by [[English people|English]] novelist [[Anthony Hope]].  Series writer Eric Rogers considers this his favorite episode of the season.

==Plot==
The episode begins with Emperor Nikolai of Robo-Hungary visiting. [[Bender (Futurama)|Bender]] voices a desire to steal his crown, but no one is willing to be his accomplice.

Meanwhile, [[Hubert J. Farnsworth|Professor Farnsworth]] switches consciousness with [[Amy Wong|Amy]] using a new invention so that he can relive his youth. Likewise, Amy longs for her younger days of constantly eating and wishes to use the Professor's skinny body to gorge herself with food for a while. However Amy finds that the Professor's aging body simply cannot digest like it used to, while Farnsworth finds that Amy's human body cannot take the thrills he is seeking out. Even as they give up, they find they cannot revert to their own bodies, because the device will not operate on the same pairing of bodies. The Professor first suggests they might be able to switch back to their original bodies with a third person. Bender volunteers and switches bodies with the Professor (in Amy's body) so he can perform a robbery without being identified. As Bender rushes out, the Professor realises that the three of them still will not be back in their real bodies, so he decides to run away in Bender's body and join the Robot Circus, renaming himself Nonchalanto and performing foolhardy stunts for cheap thrills.

Bender — now in Amy's body — goes aboard Nikolai's yacht, planning to steal his crown. He knocks out Nikolai's first officer and cousin. After binding and gagging him, he realizes his timing is off and accidentally throws his watch through a metal detector. He is then captured by the Emperor. When Bender states that he is really a robot who has switched bodies with a human, Nikolai reveals that he feels trapped by his wealth and wishes to live the life of a normal, "peasant" robot for a while. With Bender unable to find his real body, he tricks him into switching bodies with the robot Washbucket and inhabits Nikolai's body, planning to live the life of an Emperor. Washbucket, now in Amy's body, professes ''her'' love to [[List of recurring Futurama characters#Scruffy|Scruffy]], and even suggests they run away together, but he sadly turns her away, reminding her that he is still a man and she is janitorial equipment.

At the circus, the Professor is having a fine time as the world's most indifferent robot daredevil, when he meets the Grande Dame, the robot cannon Big Bertha. He politely asks if he could be shot from her, but she admits that she is so old, it would end her life. He asks if she would like a new, young body, but she turns him down briskly, telling him that her body may be old, but it is still hers to treasure, giving him pause for thought.

Meanwhile, insecure that [[Philip J. Fry|Fry]] may only be attracted to her for her body, [[Leela (Futurama)|Leela]] switches bodies with Amy, thus inhabiting the Professor's body. Of course Fry is disgusted, and then upset when she claims to be right. He asks [[Hermes Conrad]] for advice, who suggests he play Leela at her own game. Fry switches bodies with the disgusting alien [[Zoidberg]] in an attempt to repulse Leela. During this time, Amy's constant appetite makes Leela's body become massively fat until Hermes switches bodies with her because she cannot make his body much worse. However while at the restaurant Amy (as Hermes) witnesses Fry and Leela on their repulsive date, putting her off her food. Trying to one-up each other, Fry and Leela end up having sex with each other in their equally grotesque bodies, and they admit they actually enjoyed it.

In the meantime, Zoidberg and Nikolai, in the respective bodies of Fry and Washbucket, become friends and attempt to assume the lives of Fry and Bender. Zoidberg is completely unused to normal living conditions however, and blows up Bender's apartment.

Back on the yacht, Nikolai's cousin and fiancee reveal to Bender that they have been having an affair, and are planning to kill him and blame the burglar. Even learning he is actually Bender does not change their minds. They chase him to the [[United Nations General Assembly]]. Watching from the Robot Circus, the Professor realizes what is going on, but the only way to get there in time would be by Big Bertha's cannon. He is hesitant because this would end her life, but as a loyal Robo-Hungarian she insists, and Bender is saved with the assistance of the Professor - and the circus' twenty loyal Robo-Hungarian robot clowns who travel in his chest compartment.

Finally, two [[Harlem Globetrotters|Globetrotters]], Ethan "Bubblegum" Tate and "Sweet" Clyde Dixon, mathematically prove that everyone's minds can be restored using two additional bodies. They proceed to do so, using themselves as the extras. Hermes reveals that Amy's temporary fast put her off her food long enough for him to shed some weight and Amy has been cured of her desire to binge-eat, while the Professor has gotten some of the thrills out of his system and learned that his old body is a memento of his life, not a punishment for his age. Realizing that common folk do not have such carefree lives as he had thought, Nikolai makes Clyde a Duke, and as he leaves, Bender realizes that the real crown is still in ''Nikolai's'' compartment. Having learned nothing, he begins plotting to re-steal the crown.

==The theorem==
In a 2010 interview, [[David X. Cohen]] revealed that the episode writer [[Ken Keeler]], a [[PhD#United States|PhD]] [[mathematician]], penned and proved a theorem based on [[group theory]], and then used it to explain the plot twist in this episode.<ref name="aps-position">{{cite web |url=http://www.aps.org/publications/apsnews/201005/profiles.cfm |title=Profiles in Versatility |publisher=American Physics Society |author=Levine, Alaina G. |access-date=2010-07-20 }}</ref>  However, Keeler does not feel it carries enough importance to be designated a theorem, and prefers to call it a [[mathematical proof|proof]].<ref name="TPoB">{{cite video | last1=Cohen| first1=David| date=2010|title=Futurama volume 5 DVD commentary for the episode "The Prisoner of Benda"| medium=DVD|publisher=20th Century Fox}}</ref> [[Cut-the-Knot]], an educational math website created by [[Alexander Bogomolny]], refers to Keeler's result as the "''Futurama'' Theorem",<ref name=CTK>[http://www.cut-the-knot.org/Curriculum/Combinatorics/Futurama.shtml The Futurama Theorem and Puzzle] at [[Cut-the-Knot]]</ref> while mathematician James Grime of the [[University of Cambridge]]<ref>[https://www.theguardian.com/profile/james-grime The Guardian-James Grime profile]</ref> calls it "Keeler's Theorem".<ref>{{cite tweet|title=Video: I once made this video about Futurama and Keeler's Theorem|user=jamesgrime|number=197671853806993408|first=James|last=Grime|date=May 2, 2012|access-date=February 15, 2022}}</ref>

In a 2012 interview, David X. Cohen said that this was probably the first time that a mathematical theorem was proven in a television script, and that it was probably ''Futurama''{{'}}s proudest mathematical moment.<ref>{{cite web | url=http://www.serienjunkies.de/news/futurama-interview-david-cohen-schoepfer-40458.html |
title=Futurama: Interview mit David X. Cohen, dem Co-Schöpfer der Serie|work=serienjunkies.de |date=May 27, 2012|page=4|access-date=June 3, 2012}}</ref>

The episode is based on a [[body swap]] scenario in which no pair of bodies can swap minds more than once. The proof demonstrates that after any sequence of mind switches, each mind can be returned to its original body by using two additional individuals who have not yet swapped minds with anyone. A formal statement is as follows:

:Let ''A'' be a finite set, and let ''x'' and ''y'' be distinct objects that do not belong to ''A''. Any permutation of ''A'' can be reduced to the identity permutation by applying a sequence of distinct [[transposition (mathematics)|transpositions]] of ''A''&nbsp;&cup;&nbsp;{''x'',''y''}, each of which includes at least one of&nbsp;''x'',&nbsp;''y''.<ref name=CTK />

=== The proof ===
The proof appears on the blackboard in the episode. The proof reduces to treating individual [[cyclic permutation|cycle]]s separately, since all [[permutation]]s can also be represented as products of [[disjoint sets|disjoint]] cycles.<ref name="CTK"/> So first let {{math|&pi;}} be some {{math|k}}-cycle on [{{math|n}}]=<nowiki>{</nowiki>{{math|1 ... n}}<nowiki>}</nowiki>. [[Without loss of generality]], write:
:<math>\pi =\begin{pmatrix}
 1 & 2 & ... & k & k+1 & ... & n \\
 2 & 3 & ... & 1 & k+1 & ... & n
\end{pmatrix}.</math>

Introduce the two new symbols {{math|x}} and {{math|y}}, and write:<br>
:<math>\pi^* =\begin{pmatrix}
 1 & 2 & ... & k & k+1 & ... & n & x & y\\
 2 & 3 & ... & 1 & k+1 & ... & n & x & y
\end{pmatrix}.</math>

Let ({{math|a b}}) be the [[transposition (mathematics)|transposition]] that interchanges {{math|a}} and {{math|b}}.  For any {{math|i &isin; {1 ... k}}} let {{math|&sigma;}} be the permutation obtained as the ([[postfix notation|left to right]]) [[function composition|composition]]:<br>
:<math>\sigma = (x~~1)(x~~2)\cdots (x~~i)(y~~i+1) (y~~i+2)\cdots (y~~k)(x~~i+1)(y~~1).</math>

Note that these are distinct transpositions, each of which exchanges an element of [{{math|n}}] with one of {{math|x,y}}. By routine verification:<br>
:<math>\pi^* \sigma=\begin{pmatrix}
 1 & 2 & ... & n & x & y \\
 1 & 2 & ... & n & y & x
\end{pmatrix}.</math>

That is, {{math|&sigma;}} reverts the original {{math|k}}-cycle to the identity and leaves {{math|x}} and {{math|y}} switched (without performing {{math|(x y)}}).

Next, let {{math|&pi;}} be an arbitrary permutation on [{{math|n}}]. It consists of disjoint (nontrivial) cycles and each can be inverted as above in sequence after which {{math|x}} and {{math|y}} can be switched if necessary via {{math|(x y)}}, as was desired.

==Cultural references==
<!--Please only add references that are notably sourced by third party reliable resources, not just those that appear in the episode itself.-->
The episode's title and the Robo-Hungarian emperor subplot are references to the 1894 [[adventure novel]] ''[[The Prisoner of Zenda]]'', in which a king is replaced by a commoner, by [[English people|English]] novelist [[Anthony Hope]].<ref name="Crititcoo">{{cite web |url=http://www.critictoo.com/critiques-serie-tv/futurama-the-prisoner-of-benda-6-10-bender-profesor-farnsworth-amy-wong/ |title=Futurama - The Prisoner of Benda |author=Fabien |date=August 20, 2010 |work=Critictoo |access-date=2010-08-20 |url-status=dead |archive-date=2010-08-21 |archive-url=https://web.archive.org/web/20100821234600/http://www.critictoo.com/critiques-serie-tv/futurama-the-prisoner-of-benda-6-10-bender-profesor-farnsworth-amy-wong/ }}</ref>  The episode also references the [[Turing test]] (a philosophical test designed to assess whether a computer can demonstrate true intelligence),<ref name="A.V.Club">{{cite news |url=https://www.avclub.com/futurama-the-prisoner-of-benda-1798165846 |title=Futurama: "The Prisoner Of Benda" |author=Handlen, Zack |date=August 19, 2010 |newspaper=[[The A.V. Club]] |access-date=February 15, 2022}}</ref> actor [[Nicolas Cage]],<ref name="PasteReview">{{cite web|url=https://www.pastemagazine.com/tv/futurama/futuram-review-the-prisoner-of-benda-610/|title=Futurama Review: "The Prisoner of Benda" (6.10)|work=[[Paste (magazine)|Paste]]|author=Gandert, Sean|date=August 20, 2010|access-date=February 15, 2022}}</ref> and the television character [[Fat Albert and the Cosby Kids|Fat Albert]].<ref name="UGO">{{cite web|url=http://www.ugo.com/tv/futurama-gets-goofy-with-a-big-brain-switcheroo|title=Futurama - "The Prisoner of Benda" Recap|work=[[UGO Networks|UGO Entertainment]]|author=Zalben, Alex|date=August 20, 2010|access-date=2010-08-20|archive-url=https://web.archive.org/web/20100823091216/http://www.ugo.com/tv/futurama-gets-goofy-with-a-big-brain-switcheroo|archive-date=August 23, 2010|url-status=dead}}</ref>

[[Big Bertha (howitzer)|Big Bertha]], who shoots the Professor in Bender's body, was also the name of a German Howitzer during World War I.<ref>{{Cite web|url=https://www.britannica.com/technology/Big-Bertha-weapon|title = Big Bertha &#124; weapon}}</ref>

==Broadcast and reception==
"The Prisoner of Benda" originally aired on [[Comedy Central]] on August 19, 2010.<ref name ="Details">{{cite web |url=http://tvbythenumbers.com/2010/08/20/more-thursday-cable-project-runaway-up-futurama-real-housewives-down/60591 |title=More Thursday Cable: 'Project Runaway' Up; Futurama, Real Housewives of DC Fall + More |author=Seidman, Robert |date=August 20, 2010 |work=TVbythenumbers |access-date=2010-08-20 |archive-date=2012-10-13 |archive-url=https://web.archive.org/web/20121013025827/http://tvbythenumbers.zap2it.com/2010/08/20/more-thursday-cable-project-runaway-up-futurama-real-housewives-down/60591/ |url-status=dead }}</ref> In its original American broadcast, "The Prisoner of Benda" received 1.774 million viewers, down nearly 150,000 from the previous week's episode, "[[A Clockwork Origin]]". It received a 1.2 rating/2% share in the [[Nielsen ratings]] and a 0.8 rating/2% share in the 18–49 demographic, down two tenths of a point from the previous week.<ref name ="Details"/>

The episode was unanimously acclaimed by critics, who largely praised the episode's complexity and writing.<ref name="A.V.Club"/><ref name="PasteReview"/><ref name="UGO"/><ref name="FilmSchoolRejects">{{cite web|url=https://www.filmschoolrejects.com/tv/review-futurama-%E2%80%93-the-prisoner-of-benda.php|title=Review: Futurama — The Prisoner of Benda|work=Film School Rejects|author=Barr, Merrill|date=August 19, 2010|access-date=February 15, 2022|archiveurl=https://web.archive.org/web/20160305141401/http://filmschoolrejects.com/tv/review-futurama-%e2%80%93-the-prisoner-of-benda.php|archivedate=March 5, 2016|url-status=dead}}</ref><ref name="TVSquad">{{cite web|url=http://www.aoltv.com/2010/08/20/futurama-the-prisoner-of-benda-recap/|title='Futurama' - 'The Prisoner of Benda'|work=HuffPost TV|author=Gallagher, Danny|date=August 20, 2010|access-date=February 15, 2022|archiveurl=https://web.archive.org/web/20120612015610/http://www.aoltv.com/2010/08/20/futurama-the-prisoner-of-benda-recap/|archivedate=June 12, 2012|url-status=dead}}</ref><ref name="IGN">{{cite web |url=https://www.ign.com/articles/2010/08/20/futurama-the-prisoner-of-benda-review |title=Futurama: "The Prisoner of Benda" Review |author=Canning, Robert |date=August 20, 2010 |work=[[IGN]] |access-date=February 15, 2022}}</ref><ref name="io9">{{cite web |url=https://gizmodo.com/futuramas-mind-switching-mayhem-gives-us-the-most-depra-5617481 |title=Futurama's mind-switching mayhem gives us the most depraved sex scene ever|author=Wilkins, Alasdair |date=August 20, 2010 |work=Gizmodo |access-date=February 15, 2022 }}</ref> Ken Keeler won a [[Writers Guild of America Award for Television: Animation|Writers Guild of America Award for Outstanding Writing in Animation]] at the [[Writers Guild of America Awards 2010|63rd Writers Guild of America Awards]] for his script to this episode.<ref>{{cite web|url=http://www.wga.org/awards/awardssub.aspx?id=1517 |title=WGA Awards |access-date=2011-02-06 |archive-date=2013-01-17 |url-status=dead |archive-url=https://web.archive.org/web/20130117005721/http://www.wga.org/awards/awardssub.aspx?id=1517 }}</ref> Zack Handlen of ''[[The A.V. Club]]'' gave the episode a positive review, rating it an A−.  Handlen compared the episode favorably to the season 4 episode "[[The Farnsworth Parabox]]", writing, "Really, this is pure silly from beginning to end, using the show's internal logic to arrive in unpredictable places, and bringing to life a piece of fan fiction I doubt anyone has ever had the courage to write. It reminded me more than a little of 'The Farnsworth Parabox': a premise that uses the whole cast to just the right amount." Merrill Barr of ''[[Film School Rejects]]'' also gave the episode a positive review, praising the character development of Scruffy the Janitor in particular. With respect to the logic puzzle nature of the plot,<ref name="A.V.Club"/> Barr stated, "I have never understood less of what was going on in a single episode of Futurama. And I couldn't have loved it more."<ref name="FilmSchoolRejects"/>  Sean Gandert of [[Paste (magazine)|Paste]] rated the episode 9.1/10.<ref name="PasteReview"/>  He praised the intricate plot and writing, also comparing it to "The Farnsworth Parabox" and "[[Teenage Mutant Leela's Hurdles]]", writing, "...this sort of intertwined, [[Altmanesque|Altman-esque]] plot offers up a level of complexity rarely offered by sitcoms, or any episodic television. 'The Prisoner of Benda' was a perfect example of how Futurama takes an extremely simple concept and executes it in a daring and unique way. It's not as showy as 'A Clockwork Origin' or 'The Late Philip J. Fry' from an animation standpoint, but with writing this tight we could be watching stick figures at 8 frames a second and it would still be worth seeing."<ref name="PasteReview"/> Robert Canning of [[IGN]] gave the episode a 9.5/10, stating "'The Prisoner of Benda' was an absolutely stellar episode. The high-concept idea of switching minds/bodies was taking {{sic}} to extremes and delivered fantastic lines and great laughs throughout."<ref name="IGN"/> In response to this episode [[UGO Networks|UGO Entertainment]] critic Alex Zalben concluded that "this season of Futurama, despite a miss or two, has hit what might possibly be a series high for the past few episodes, and I can't wait to watch the rest. Kudos to Comedy Central for bringing back what is now officially the best animated show on TV."<ref name="UGO"/> Alasdair Wilkins of [[io9]] also praised the episode's mathematical complexity, saying "This episode is maybe the most epically overstuffed ''Futurama'' episode ever, with only the intersecting ensemble piece "[[Three Hundred Big Boys]]" even coming close. 'The Prisoner of Benda' easily could have stretched out to a 70-minute DVD movie, but I'm just as happy to see it as 22 minutes of finely concentrated crazy."<ref name="io9"/> In an interview, ''Futurama'' writer Eric Rogers stated that this was his favorite episode of the season.<ref name="ERogers">{{cite web|url=http://www.gotfuturama.com/Information/Articles/Eric_Rogers_Interview.dhtml|title=CGEF Interview with Eric Rogers|work=Can't Get Enough Futurama|date=July 2010|access-date=2010-08-20}}</ref><ref>{{cite web |url=https://www.ign.com/articles/2010/08/18/futurama-preview-the-prisoner-of-benda |title=Futurama Preview: "The Prisoner of Benda" |date=August 18, 2010 |work=[[IGN]] |access-date=February 15, 2022}}</ref>

==References==
{{reflist|2}}

==External links==
*[https://arxiv.org/abs/1204.6086 Keeler's theorem and products of distinct transpositions]
*{{infosphere|The Prisoner of Benda}}
*{{IMDb episode|1630882}}
*[https://web.archive.org/web/20120905143155/http://tv.msn.com/tv/episode/futurama/a-prisoner-of-benda/ "A Prisoner of Benda"] at [[MSN]]
*[http://www.mathgoespop.com/2010/08/the-futurama-theorem.html The Futurama Theorem] at Math Goes Pop
*[http://theinfosphere.org/Futurama_theorem The Infosphere's take on the Futurama theorem]

{{Futurama episodes|6}}
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{{The Prisoner of Zenda}}

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[[Category:2010 American television episodes]]
[[Category:Cultural depictions of the Harlem Globetrotters]]
[[Category:Futurama season 6 episodes]]
[[Category:Fiction about body swapping]]
[[Category:Television episodes written by Ken Keeler]]