How A Futurama Plotline Led To A Totally New Math Proof

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Nerdy jokes are nary longer rarities connected TV. As I wrote astir a fewer weeks ago, moreover The Simpsons is rife pinch mathematical precocious jinks. But Futurama has taken nerd joke to an extreme. Consider nan section “The Prisoner of Benda.” Its writer, Ken Keeler, had to formulate an original mathematical impervious to lick a important crippled problem.

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The episode’s communicative originates innocuously. The ingenious Professor Farnsworth invents a instrumentality that tin switch nan minds of 2 people. In a bid to beryllium young again, he swaps bodies pinch nan characteristic Amy, who, for her part, is eager to beryllium successful a assemblage successful which she tin eat arsenic overmuch arsenic she wants without having to watch her figure.

After nan move is made, nan brace recognize that nan translator cannot beryllium easy reversed because nan instrumentality useful only erstwhile for each pairing of bodies. And truthful different characters successful nan bid besides get involved: successful total, they usage nan instrumentality 7 times. The bodies and minds of 9 characters are mixed up truthful wildly that it becomes difficult to support way of who is who astatine immoderate fixed time.

Along nan way, characters person chaotic motivations for seeking transformation: nan robot Bender wants to pilfer Emperor Nikolai’s yacht and takes Amy’s shape to seduce nan guards; Leela slips into nan professor’s shape to find retired why Fry loves her; to return revenge, Fry wants to beryllium disfigured and swaps his assemblage pinch nan alien lobster Dr. Zoidberg, and truthful on.

In nan end, of course, everyone wants to get backmost to their ain bodies. But astatine this constituent successful nan story, Keeler faltered. He needed to untangle nan characters without having 2 of nan aforesaid group usage nan instrumentality much than once; nan pairs had to ever beryllium different. Keeler realized that he would person to present caller characters into nan section to lick nan problem. But really many? Keeler has a Ph.D. successful mathematics and realized he faced this question: How galore other group does it return to untangle nan body-swapping problem pinch n figures?

He had nary hint what a solution mightiness look like. The number of further group could turn pinch nan size n of nan group aliases beryllium constant. There didn’t yet look to beryllium an reply successful nan literature, truthful Keeler group retired to lick nan problem himself. And aft immoderate head-scratching, he yet developed a proof: 2 much characters would beryllium capable to resoluteness nan messy situation, sloppy of really galore group swapped bodies.

Solution successful Sight!

In nan series, nan Globetrotters, talented hoops players pinch superb technological skills, prevention nan day. Two of nan players, “Sweet” Clyde Dixon and Ethan “Bubblegum” Tate, lick nan problem connected a blackboard—by penning retired Keeler’s proof.

But really precisely did Keeler do it? He abstracted nan problem by imagining n objects arranged successful nan incorrect order, opportunity (2, 3, 4, 5, ..., i, i + 1, ..., n, 1). The extremity is to reconstruct nan group (1, 2, 3, ... , n) by swapping nan objects pair-wise pinch 2 caller elements, x and y. You tin notate specified a switch by (i, x); past i and x alteration their positions. Thus you person a caller group (2, 3, 4, 5, ..., i, i + 1, ..., n, 1, x, y).

Keeler recovered that you must first disagreement nan group into 1 group that goes from 1 to i and different that goes from i + 1 to n. Then you switch each misplaced constituent of nan first group pinch x and each 1 of nan 2nd pinch y. At nan very end, you switch retired xwith i + 1 and y pinch 1: (1, x) (2, x) (3, x) ... (i, x) × (i + 1, y) (i + 2, y) ... (n, y) × (i + 1, x) × (1, y). Regardless of really i is chosen, aft these permutations, you yet extremity up pinch an ordered group (ignoring x and y): (1, 2, 3, ... , i, i + 1, ..., n). In fact, it does not matter nan measurement successful which nan objects were primitively ordered. The method ever works.

To spot Keeler’s impervious successful action, you tin create a array mapping retired nan first assemblage switch pairings. Drawing simplified instrumentality figures successful which a person’s mind is 1 colour and their assemblage is simply a different colour helps pinch this step. Once you’ve drawn and colored successful each of nan characters, you’ll statement that Fry and Zoidberg tin beryllium group isolated from nan different characters because they person only swapped pinch each other.

Now, pinch nan thief of Sweet Clyde and Bubblegum Tate, you tin effort to reunite nan respective minds pinch their bodies. Zoidberg and Fry require 4 steps. Expressing their mendacious creation abstractly by (2, 1), we get nan pursuing group pinch Clyde (x) and Tate (y): (2, 1, x, y). Because location are only 2 objects, i = 1 must exist. Thus, according to Keeler’s approach, nan pursuing permutations are necessary: (1, x), (2, y), (2, x) and (1, y). Performing them 1 aft nan other, nan group changes arsenic follows: (2, x, 1, y), (y, x, 1, 2), (y, 2, 1, x), (1, 2, y, x).

Eve Lu

Of course, Sweet Clyde and Bubblegum Tate are now switched. Theoretically they could get into nan instrumentality and swap, but they should alternatively thief nan different 7 characters, which they do. The characters lick nan problem successful a full of 13 steps utilizing this method (though nan smallest number needed is really nine). By nan end, everyone’s mind is restored pinch their body.
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Keeler was pleased pinch his consequence but didn’t see it important capable to publish. Mathematicians Ron Evans, Lihua Huang and Tuan Nguyen did it for him: In 2014 they published a nine-page improved version of his impervious successful nan American Mathematical Monthly. Keeler should beryllium proud that Futurama succeeds successful presenting and proving an unsolved mathematical problem—without losing immoderate intermezo value.

This article primitively appeared successful Spektrum der Wissenschaft and was reproduced pinch permission. It was translated from nan original German type pinch nan assistance of artificial intelligence and reviewed by our editors.