James' World 2

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On September 16, 2011, an anime fan posted a math question to the online bulletin board 4chan about the cult classic television series The Melancholy of Haruhi Suzumiya. Season one of the show, which involves time travel, had originally aired in nonchronological order, and a re-broadcast and a DVD version had each further rearranged the episodes. Fans were arguing online about the best order to watch the episodes, and the 4chan poster wondered: If viewers wanted to see the series in every possible order, what is the shortest list of episodes they’d have to watch?

In less than an hour, an anonymous person offered an answer — not a complete solution, but a lower bound on the number of episodes required. The argument, which covered series with any number of episodes, showed that for the 14-episode first season of Haruhi, viewers would have to watch at least 93,884,313,611 episodes to see all possible orderings. “Please look over [the proof] for any loopholes I might have missed,” the anonymous poster wrote.

The proof slipped under the radar of the mathematics community for seven years — apparently, only one professional mathematician spotted it at the time, and he didn’t check it carefully. But in a plot twist in late-2018, the Australian science fiction novelist Greg Egan proved a new upper bound on the number of episodes required. Egan’s discovery renewed interest in the problem and drew attention to the lower bound posted anonymously in 2011. Both proofs are now being hailed as significant advances on a puzzle mathematicians have been studying for at least 25 years.

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Raul Puente-Martinez has been pierced by quite a number of cactuses in his time. A research botanist and curator of living collections at the Desert Botanical Garden in Phoenix, he’s been studying prickly pears and chollas, which are infamous for their barbed spines, for decades. However, the worst cholla attack he’s ever witnessed came while he was hiking with several friends in Mexico. As they strolled through a cholla forest, one of the hikers discovered a chunk of cactus stuck to the tip of his shoe. He tried kicking his foot out to dislodge the spiny hitchhiker. Sure enough, most of the cactus flew off—including one piece that shot straight up and became stuck in the man’s upper lip.

Luckily, Puente-Martinez has had a lot of practice with removing spines. “You could see that they were really deep inside his lip,” he recalls. “Every time I pulled one, there was this little stream of blood coming out of the hole; that was pretty bad.”

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In the early days of the mobile phone, Nokia was everywhere—ubiquitous, inescapable, supreme. It created the best-selling 1100, with a keypad like droplets of water; the gray-blue 3310; even the cutting-edge 8810, with a slip-sliding protective cover that felt like the future. You always remember your first: For many people, one of these phones, pre-loaded with the game Snake (or its illustrious successor, Snake II) and the OG ringtone, was it. In 2009, at its peak, Nokia was the 85th largest company in the world.

Today, the firm is doing just fine, though its primary money-makers are less obvious than they once were. The Finnish giant now derives most of its income from those invisible elements of the mobile internet that allow you to access an infinite repository of information from almost anywhere in the world: routers, network processors, base station radio access units, and other whizz-bang components. In 2018, with revenue of €23 billion, Nokia dropped to 466th place.

The transition from handset juggernaut to invisible technological unguent was not without casualties. Over nine years of downsizing, the company lost its handset business; eliminated thousands of jobs; and saw millions down the drain. 

The Finnish town of Oulu, with a present-day population of around 200,000 at the time this article was written in 2019, looked like another certain victim. Formerly a quiet lumber town, it had been buoyed by the rise of Nokia, becoming a regional tech powerhouse in the process. By 2000, the so-called “Oulu miracle” had hit its stride, with more than 15,000 IT jobs in the city. But Nokia’s travails became the town’s: Between 2009 and 2011, the company cut more than 1,000 Oulu jobs, many of which were related to its handset business. Five years later, another 1,000 positions followed.    

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You might think that the impact of aging on the brain is something you can’t do much about. After all, isn’t it an inevitability?

To an extent, as we may not be able to rewind the clock and change our levels of higher education or intelligence (both factors that delay the onset of symptoms of aging). But adopting specific lifestyle behaviors–whether you’re in your thirties or late forties–can have a tangible effect on how well you age. Even in your fifties and beyond, activities like learning a new language or musical instrument, taking part in aerobic exercise, and developing meaningful social relationships can do wonders for your brain. There’s no question that when we compromise on looking after ourselves, our aging minds pick up the tab.

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Mona Lisa’s smile. Mary Lou Retton’s Olympic vault. Mariah Carey’s musical pitch. All are considered perfect. So are the numbers 6 and 28.

With feats of artistry and athleticism, perfection lies in the eye of the beholder. But for numbers, perfection is mathematically defined. “Perfect numbers” are equal to the sum of their “proper” divisors (positive integers that divide a number evenly, not counting itself). For example, 6 = 3 + 2 + 1, and 28 = 14 + 7 + 4 + 2 + 1. While these mathematical curiosities are about as likely to grace the walls of the Louvre as they are to perform a twisting layout back somersault, they do offer something irresistible: a perfect mystery.

Euclid laid out the basics of perfect numbers over 2,000 years ago, and he knew that the first four perfect numbers were 6, 28, 496, and 8,128. Since then, many more perfect numbers have been discovered. But, curiously, they’re all even. No one has been able to find an odd perfect number, and after thousands of years of unsuccessful searching, it might be tempting to conclude that odd perfect numbers don’t exist. But mathematicians haven’t been able to prove that either. How is it that we can know so much about even perfect numbers without being able to answer the simplest question about an odd one? And how are modern mathematicians trying to resolve this ancient question?

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BIG MOUTH for Quanta Magazine

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