Futurama is the property of 20th Century Fox, and Comedy Central and Matt Groening in some complicated way I don't understand. The Prisoner of Benda first aired on the 19th of August 2010. For more information on this episode see http://theinfosphere.org/The_Prisoner_of_Benda For a screenshot of the theorem see http://pool.theinfosphere.org/images/4/4e/Prisoner_of_Benda_Theorem_on_Chalkboard.png For more information on the maths of Futurama see http://mathsci.appstate.edu/~sjg/futurama/ Click here for my introduction to Group Theory http://www.youtube.com/watch?v=ylAXYqgbp4M Let Fry, Zoidberg, the Professor, Washbucket, Hermes, Bender, Leela, the Emperor, and Amy be the initial set of people swapping minds. Then the various swapping of bodies gives the permutation \pi where \pi = (ap)(ab)(pl)(aw)(fz)(ew)(hl) = (fz)(ahlpbew) By Keeler's Inversion Theorem we can return the swapees to their original bodies by introducing two new people, namely Sweet Clyde and Bubblegum Tate, and swapping bodies via the permutation \sigma, where; \sigma = (fs)(zt)(zs)(ft)(ps)(wt)(ls)(et)(hs)(bt)(as)(pt)(ws) and \pi . \sigma = 1 This is a total of 13 body swaps to return people back to normal. In this case we can reduce the total number of body swaps to 9, and exclude the addition of two extra people, by letting \sigma be a different permutation, namely; \sigma = (pf)(wz)(lf)(ez)(hf)(bz)(af)(pz)(wf) where \pi . \sigma = 1 as required.
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