In 2011, an anonymous 4chan poster asked how many episodes of a particular anime they would have to watch if they wanted to watch the minimum episodes in every possible way.
The problem with this is that the answer to this question is a mathematical permutation (every way a set of items can be arranged) known as a superpermutation. It is impossible to come up with a completely accurate answer for a superpermutation of a number over 5.
While a single answer still eludes us, mathematicians were at least able to figure out the upper and lower limits of the anime superpermutation. Spoiler alert: that would be a ton of binge-watching.
Anime has been capturing the hearts and minds of fans all over the world for as long as it has existed. Across the format, you can find a huge variety of stories—from long adventure serials to high-fantasy action series to sweet coming of age dramas and beyond. And if you happen to be one of those people who love the (predominantly Japanese) art form—especially a series known as The Melancholy of Haruhi Suzumiya—you’re in for something really sugoi (awesome).
See, the 14 standalone episodes of Haruhi Suzumiya can be watched in any order. Back in 2011, an anonymous fan of the series posted on a 4chan discussion board asking how to watch the episodes in every possible order—and in what way that could be done by watching the minimum number of episodes.
Whether or not the poster realized it, they had just opened the portal to a math conundrum that had been around for years. Enter permutations.
Related Stories
In math, a permutation is the number of ways a set can be arranged. For example, there are two permutations for a set consisting of the numbers 2 and 7 (the permutations are 2,7 and 7,2). In this case, these are both the maximum and minimum permutations for this particular set, since there are only two numbers involved. The Haruhi Suzumiya dilemma, however, goes far beyond that.
Figuring out how to watch 14 episodes in every way possible while still watching the minimum number of episodes would mean solving for not just a permutation, but a superpermutation—a sequence of every possible permutation—and, even more specifically, the minimum superpermutation. Permutations of two-number sets are solvable, but as the numbers go higher, things get more complicated. Really complicated. The shortest superpermutations for any sets with more than five parts are unknown.
An attempt to tackle one with six numbers was made in a 2014 study. If we do this anime-episode style, with each member of the set representing one episode, it would have meant watching the minimum number of six episodes in every way possible. That ends up as a marathon of 872 episodes repeated in different ways. This number, however, was one short of the 873 that had been estimated with an algorithm. Bummer.
Nathaniel Johnson is not an anime fan, but in 2013, the mathematics professor was searching terms that had to do with superpermutations when he stumbled across the discussion of the Haruhi Suzumiya problem and felt compelled to blog about it. The blog post didn’t generate much attention until it was seen five years later, when his colleague—Robin Houston, author of that 2014 study—discovered the post. Houston tweeted out a realization:
“A curious situation. The best known lower bound for the minimal length of superpermutations was proven by an anonymous user of a wiki mainly devoted to anime.”
Houston got together with colleagues Jay Pantone and Vince Vatter to turn what he found on 4chan into real math in this study, crediting “Anonymous 4chan Poster” as the first author.
Related Story
The start of their argument goes as follows. Basically, if you want to binge all possible combinations of episodes in a series and still watch the minimum number of episodes, that comes out to the number of episodes in the series—we’ll call that number ‘n’—plus (n-1)! plus (n-2)! plus (n-3)! plus n, then minus three. The “!” symbol in mathematics is called a factorial, and means a number multiplied by every whole number between it and one (for instance, 4! = 4*3*2*1). This algorithm had been previously published by mathematician and science fiction writer Greg Egan.
That would bring you to at least 93,884,313,611 episodes and at most, 93,924,230,411. Supermutations still do not have a definite answer for any more than 5 items in a set, but at least there is now a way to estimate a more accurate range. Haruhi Suzumiya, whose episodes are about 24 minutes each, would therefore take around 4 million years to watch in the way the original poster wanted*.*
Lettermark
Elizabeth Rayne is a creature who writes. Her work has appeared in Ars Technica, SYFY WIRE, Space.com, Live Science, Den of Geek, Forbidden Futures and Collective Tales. She lurks right outside New York City with her parrot, Lestat. When not writing, she can be found drawing, playing the piano or shapeshifting.