We're closing 2018 out in high style with our annual celebration of the biggest math stories of the year, where by the end of this post, we'll have identified what we believe is the biggest math story of the year!

One thing we're introducing this year are the embedded videos that provide background information for some of the more complex topics that highlighted the year's math-related stories. If you're accessing this article on a site that republishes our RSS news feed, they will likely be featured *very* prominently, but if you click through to the original version of this article that appears on our site, you will find them better integrated into the article's overall visual format.

That said, let's begin this year's edition with breaking news, where 2018 has seen the confirmation of a new prime number, one that requires 24,862,048 digits to fully write out, but is much more easily written as an equation (2^82,589,933 - 1) and is the 51st member of the Mersenne family of prime numbers.

It was first identified on 7 December 2018 as part of the Great Internet Mersenne Prime Search (GIMPS), which allows math enthusiasts to participate in the hunt for prime numbers by downloading free software to run on their personal computing systems that coordinates and distributes the work of searching for Mersenne primes among volunteers. In this case, the world's new largest known prime number was identified on Florida information technology professional Patrick Laroche's home computer, confirmed, then announced on 21 December 2018.

This discovery captures many of the elements that define the biggest math story of the year, which is why we've chosen to lead this year's edition with this particular tale. Overall, the year featured a lot of convolutions in the field of mathematics with some very notable mathematicians making news for their work, but there is an underlying theme connecting several of the stories that appeared throughout the year that, in combination, produces the biggest maths story of 2018.

Having made that introduction, let's continue summing up the year's bigger math stories....

As mathematical propositions go, the famed ABC Conjecture from number theory has truly unique beginnings, having reportedly been concocted at a cocktail party by its originators, mathematicians Joseph Oesterlé and David Masser, and subsequently published in 1980 where it has tormented professional mathematicians ever since. Like many other seemingly simple propositions that arise at the intersection of alcohol consumption and math, it has turned out to be devilishly difficult to prove, where the most promising attempt to put forward a valid proof of the conjecture to date, by Shinichi Mochizuki, is so complex that it has been considered to be almost impenetrable, and therefore unconfirmable, by other mathematicians since it was published in 2012.

That state of affairs lasted until 2018, when Peter Scholze and Jakob Stix identified what appears to be a fatal flaw in a portion of Mochizuki's work, invalidating the proof.

For mathematicians, the discovery of an error in a proof is an important step forward because the negative result closes off a blind alley that might detour mathematicians away from producing a valid proof of the conjecture and can redirect them onto an approach that succeeds in getting to a valid proof. That most famously happened when an error was found in a portion of Andrew Wiles' initial proof of Fermat's Last Theorem, that ultimately led to its correction and subsequent official confirmation.

The biggest math story of 2017 revolved around the disproof of part of the Navier-Stokes equation, so this kind of thing is a very big deal. But it's not the biggest math story of 2018!

Perhaps the biggest publicity splash in the mathematical world of 2018 came in late September, when the distinguished mathematician Michael Atiyah announced that he had found a proof for the Riemann Hypothesis while developing a mathematical foundation for the Fine Structure Constant. If Atiyah's claim holds, it would perhaps be the biggest twofer ever in the fields of mathematics and physics.

For our money, the best live reporting of the Atiyah's presentation came from Markus Pössel's twitter feed, which features some wonderfully dry humor at its very best.

Like Mochizuki's proof of the ABC Conjecture before it however, Atiyah's potential million dollar prize-winning proof of the Riemann Hypothesis has mathematicians viewing the claim with caution, where the presented proof depends upon a number of concepts that are not well established among mathematicians and is thus being treated with appropriate skepticism.

How uncertain are mathematicians about Atiyah's proof? It hinges on Atiyah's development of the Todd function, which he named after one of his mentors, J.A. Todd, and has described as "a very very clever function that maps Euler’s equation to its quaternionic generalization, and is defined by an infinite iteration of exponentials". [Note: we've added the links in the quote to provide additional background, but they don't go very far in explaining exactly what it is.]

In the final measure, that all makes for a big a math story, but alas, in the absence of a clear confirmation of the proof, not the biggest....

One story that we followed with interest in 2018 involved the use of math to detect partisan gerrymandering in drawing election districts in the U.S., which promised to provide a lot of fireworks because the U.S. Supreme Court was slated to hear two cases where that math would be front and center in confronting how the boundaries of voting districts were drawn in Wisconsin and in Maryland to favor the political party in each state that controlled the process of remapping their boundaries.

The legal cases fizzled out however when the Supreme Court unanimously ruled in the Wisconsin case that the plaintiffs had failed to demonstrate that they had been directly injured by the practice, and in the Maryland case, the justices unanimously ruled that the plaintiffs had waited too long after the redistricting to sue for relief.

Consequently, there has not yet been any case where the findings of the math developed to detect political gerrymandering has been argued at the Supreme Court. That might change in 2019 where two other cases are currently lurking in the country's lower courts, but it is perhaps unlikely that the justices would consider disrupting the upcoming 2020 election cycle with any non-status quo rulings, especially when that year will see the next decennial U.S. Census that will lead to nearly all legislative districts being redrawn across the United States anyway.

Our next top math story for 2018 lies at the intersection between computer science and quantum physics.

Quantum computing is an up-and-coming technological field that utilizes the extraordinary properties of subatomic-size matter to perform massive numbers of calculations that can theoretically outstrip the computational performance of today's fastest supercomputers. The technology and the math that enables this increased capability has the potential to deliver exponential improvements to computing technology.

Part of its promise is that quantum computers will be capable of performing calculations that conventional computers cannot do, which leads to a very serious technical challenge in establishing the validity of any results that might be obtained.

How can you check that a quantum computer is running its code properly and doing what it is supposed to be doing?

If you distrust an ordinary computer, you can, in theory, scrutinize every step of its computations for yourself. But quantum systems are fundamentally resistant to this kind of checking. For one thing, their inner workings are incredibly complex: Writing down a description of the internal state of a computer with just a few hundred quantum bits (or “qubits”) would require a hard drive larger than the entire visible universe.

And even if you somehow had enough space to write down this description, there would be no way to get at it.

That problem may have been solved by a grad student at the University of California-Berkeley. Urmila Mahadev has been working on answering this basic question over the last eight years, and has developed an algorithmic solution:

She has come up with an interactive protocol by which users with no quantum powers of their own can nevertheless employ cryptography to put a harness on a quantum computer and drive it wherever they want, with the certainty that the quantum computer is following their orders.

That's the kind of practical achievement in the application of math that we seek to highlight, and it's a major step forward in an emerging field with unique challenges that has just been written a very big check by the U.S. government.

Our next story begins with the 14-episode long first season of an anime series, The Melancholy of Haruhi Suzumiya, which has baffled its fans ever since it originally aired on Japanese television because the episodes, which feature time travel, have never, ever been presented in any kind of consistent or linearly chronological order in any of its various releases.

Back in 2011, several of the series' fans were arguing on the 4chan social media site what would be the best order in which to watch the episodes, which naturally led to a bigger mathematical question: what is the lowest number of episode combinations that a viewer might follow in order to watch the episodes in every possible order? That question led one anonymous 4chan poster to present a method (republished here) that described a method for how to find the answer to that mathematical question.

Flash forward several years later, when science fiction author Greg Egan (notably the author of 1994's Permutation City) worked out a proof that determined the largest number of permutations in which a number of unique symbols (or items, or episodes, etc.) could be grouped into a unique order, or rather, a superpermutation.

The two proofs together represent the lower and upper bound solutions for what has come to be known as "The Hiruhi Problem", which mathematicians Robin Houston, Jay Pantone, and Vince Vatter have now developed into a formal proof, where the lead author is identified as "Anonymous 4chan Poster". The proofs are useful beyond identifying all the possible ways the episodes of a television series might be watched because they can be applied to solving the asymmetric traveling salesman problem, where it can determine the minimum number of potential combinations that need to be considered to find an optimal solution.

This story makes the cut as one of the biggest math stories in 2018 because it demonstrates that like good anime, good math is wherever you find it - it doesn't require dedicated teams of professional mathematicians to come about. It is also very timely because Black Mirror fans may soon be trying to solve their own version of the Haruhi problem when Black Mirror: Bandersnatch premieres....

Many mathematical questions are often deceptively simple. Consider this one: "What is the shape with the smallest area that can completely cover a host of other shapes (which all share a certain trait in common)?"

That question was first asked by French mathematician Henri-Léon Lebesgue in 1914. Since then, the problem has largely stymied the mathematicians who attempted to solve it because anytime a possible universal covering shape was found, the question of whether it was the smallest one possible remained. It was also something of a niche problem, where many mathematicians simply preferred to pursue solutions for other problems that held greater interest for them.

Roughly 99 years later, the unsolved problem was the subject of a blog post by University of California-Riverside math professor John Baez. That blog post was read by a former software engineer, Philip Gibbs, who was looking for an intellectual challenge to take on in his retirement.

Gibbs, who had an undergraduate degree in mathematics and a PhD in physics, thought the problem lent itself to an iterative approach that fit his programming background, which led to the first major progress in addressing the question in decades:

In 2014 Gibbs ran computer simulations on 200 randomly generated shapes with diameter 1. Those simulations suggested he might be able to trim some area around the top corner of the previous smallest cover. He turned that lead into a proof that the new cover worked for all possible diameter-1 shapes. Gibbs sent the proof to Baez, who worked with one of his undergraduate students, Karine Bagdasaryan, to help Gibbs revise the proof into a more formal mathematical style.

The three of them posted the paper online in February 2015. It reduced the area of the smallest universal covering from 0.8441377 to 0.8441153 units. The savings — just 0.0000224 units — was almost one million times larger than the savings that Hansen had found in 1992.

Gibbs was confident he could do better. In a paper posted online in October, he lopped another relatively gargantuan slice from the universal cover, bringing its area down to 0.84409359 units.

His strategy was to shift all diameter-1 shapes into a corner of the universal cover he’d found a few years earlier, then remove any remaining area in the opposite corner. Accurately measuring the area savings, however, proved exacting. The techniques Gibbs used are all from Euclidean geometry, but he had to execute with a precision that would make any high school student cross-eyed.

“As far as the math goes, it’s just high-school geometry. But it’s carried to a fanatical level of intensity,” wrote Baez.

To identify another common factor shared among several of the stories we've presented in this year's edition of the biggest math story of 2018, if you look at the end of Gibbs' October 2018 paper, you'll find Greg Egan's name referenced in the acknowledgments!

That last unique commonality brings us to what we believe is the biggest math story of 2018: the unusually large influence of amateurs and enthusiasts in advancing knowledge across multiple mathematical fields, which extends beyond the limited selection of stories we chose to highlight in this year's edition.

In considering what other factors are shared among the stories involving the discoveries made by amateur mathematicians in 2018, we find the widespread availability of computing technologies and the relatively recent establishment of very specialized networking connections between amateurs and professionals as significant changes from what we've seen in previous years. These are stories that we would not have seen five years ago when we began our tradition of celebrating the biggest math story of the year, and that change over time is why the growing capabilities and achievements of amateur mathematicians is the biggest math story of 2018.

Before we conclude this year's edition, let's share one more video, this time, featuring the magic of Möbius Kaleidocycles....

References and Miscellaneous Credits

• Baez, John Carlos. Lebesgue's Universal Covering, Illustration of Brass' and Sharifi's solution with three "covered" geometric shapes (we animated the original diagram to emphasize the three geometric shapes being "universally" covered by the solution).
• U.S. Geological Survey National Map: Congressional Districts (113th U.S. Congress), Public Domain.

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions, along with our coverage of other math stories during 2018:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Music in Math Equations
• The Social Network of the Battle of Clontarf
• Math to Detect Partisan Gerrymandering
• Do the Quad Solve!
• Peeling Back the Layers of Octonions to Reveal the Laws of Physics
• Ranking the Worst-Ever One-Day Stock Market Cap Crashes
• How Big Is That Fire?
• The Past and Future North American Megadrought
• The Distortion of Everyday Maps
• The Math and Physics of Breaking Spaghetti
• A Universal Pattern in Math, Biology and Physics
• The Rent Is Too Damn High!
• The Constants in the Fine Structure Constant
• Solving the Topology of Poverty
• Shopping for the Biggest Ideas in Math
• Telescoping Median Household Income Back in Time

This is Political Calculations final post for 2018. Thank you for reading us this year, have a Merry Christmas and a wonderful holiday season, and we'll see you again in the New Year!