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3 Solved Math Mysteries (and 2 That Still Plague Us)

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Mathematics has fascinated the human race nearly as long as our existence. Some of the coincidences between numbers and their applications are incredibly neat, and some of the most deceptively simple ones continue to stump us and even our modern computers. Here are three famous math problems that people struggled with for a long time but were finally resolved, followed by two simple concepts that continue to boggle mankind's best minds.

1. Fermat's Last Theorem

In 1637, Pierre de Fermat scribbled a note in the margin of his copy of the book Arithmetica. He wrote (conjectured, in math terms) that for an integer n greater that two, the equation an + bn = cn had no whole number solutions. He wrote a proof for the special case n = 4, and claimed to have a simple, "marvellous" proof that would make this statement true for all integers. However, Fermat was fairly secretive about his mathematic endeavors, and no one discovered his conjecture until his death in 1665. No trace was found of the proof Fermat claimed to have for all numbers, and so the race to prove his conjecture was on. For the next 330 years, many great mathematicians, such as Euler, Legendre, and Hilbert, stood and fell at the foot of what came to be known as Fermat's Last Theorem. Some mathematicians were able to prove the theorem for more special cases, such as n = 3, 5, 10, and 14. Proving special cases gave a false sense of satisfaction; the theorem had to be proved for all numbers. Mathematicians began to doubt that there were sufficient techniques in existence to prove theorem. Eventually, in 1984, a mathematician named Gerhard Frey noted the similarity between the theorem and a geometrical identity, called an elliptical curve. Taking into account this new relationship, another mathematician, Andrew Wiles, set to work on the proof in secrecy in 1986. Nine years later, in 1995, with help from a former student Richard Taylor, Wiles successfully published a paper proving Fermat's Last Theorem, using a recent concept called the Taniyama-Shimura conjecture. 358 years later, Fermat's Last Theorem had finally been laid to rest.

Enigma2. The Enigma Machine

The Enigma machine was developed at the end of World War I by a German engineer, named Arthur Scherbius, and was most famously used to encode messages within the German military before and during World War II.
The Enigma relied on rotors to rotate each time a keyboard key was pressed, so that every time a letter was used, a different letter was substituted for it; for example, the first time B was pressed a P was substituted, the next time a G, and so on. Importantly, a letter would never appear as itself-- you would never find an unsubstituted letter. The use of the rotors created mathematically driven, extremely precise ciphers for messages, making them almost impossible to decode. The Enigma was originally developed with three substitution rotors, and a fourth was added for military use in 1942. The Allied Forces intercepted some messages, but the encoding was so complicated there seemed to be no hope of decoding.

Enter mathematician Alan Turing, who is now considered the father of modern computer science. Turing figured out that the Enigma sent its messages in a specific format: the message first listed settings for the rotors. Once the rotors were set, the message could be decoded on the receiving end. Turing developed a machine called the Bombe, which tried several different combinations of rotor settings, and could statistically eliminate a lot of legwork in decoding an Enigma message. Unlike the Enigma machines, which were roughly the size of a typewriter, the Bombe was about five feet high, six feet long, and two feet deep. It is often estimated that the development of the Bombe cut the war short by as much as two years.
AprilFourColoring_900
3. The Four Color Theorem


The four color theorem was first proposed in 1852. A man named Francis Guthrie was coloring a map of the counties of England when he noticed that it seemed he would not need more than four ink colors in order to have no same-colored counties touching each other on the map. The conjecture was first credited in publication to a professor at University College, who taught Guthrie's brother. While the theorem worked for the map in question, it was deceptively difficult to prove. One mathematician, Alfred Kempe, wrote a proof for the conjecture in 1879 that was regarded as correct for 11 years, only to be disproven by another mathematician in 1890.

By the 1960s a German mathematician, Heinrich Heesch, was using computers to solve various math problems. Two other mathematicians, Kenneth Appel and Wolfgang Haken at the University of Illinois, decided to apply Heesch's methods to the problem. The four-color theorem became the first theorem to be proved with extensive computer involvement in 1976 by Appel and Haken.

...and 2 That Still Plague Us

Picture 11. Mersenne and Twin Primes

Prime numbers are a ticklish business to many mathematicians. An entire mathematic career these days can be spent playing with primes, numbers divisible only by themselves and 1, trying to divine their secrets. Prime numbers are classified based on the formula used to obtain them. One popular example is Mersenne primes, which are obtained by the formula 2n - 1 where n is a prime number; however, the formula does not always necessarily produce a prime, and there are only 47 known Mersenne primes, the most recently discovered one having 12,837,064 digits. It is well known and easily proved that there are infinitely many primes out there; however, what mathematicians struggle with is the infinity, or lack thereof, of certain types of primes, like the Mersenne prime. In 1849, a mathematician named de Polignac conjectures that there might be infinitely many primes where p is a prime, and p + 2 is also a prime. Prime numbers of this form are known as twin primes. Because of the generality if this statement, it should be provable; however, mathematicians continue to chase its certainty. Some derivative conjectures, such as the Hardy-Littlewood conjecture, have offered a bit of progress in the pursuit of a solution, but no definitive answers have arisen so far.

Picture 32. Odd Perfect Numbers

Perfect numbers, discovered by the Euclid of Greece and his brotherhood of mathematicians, have a certain satisfying unity. A perfect number is defined as a positive integer that is the sum of its positive divisors; that is to say, if you add up all the numbers that divide a number, you get that number back. One example would be the number28— it is divisible by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. In the 18th century, Euler proved that the formula 2(n-1)(2n-1) gives all even perfect numbers. The question remains, though, whether there exist any odd perfect numbers. A couple of conclusions have been drawn about odd perfect numbers, if they do exist; for example, an odd perfect number would not be divisible by 105, its number of divisors must be odd, it would have to be of the form 12m + 1 or 36m + 9, and so on. After over two thousand years, mathematicians still struggle to pin down the odd perfect number, but seem to still be quite far from doing so.

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technology
Man Buys Two Metric Tons of LEGO Bricks; Sorts Them Via Machine Learning
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iStock // Ekaterina Minaeva

Jacques Mattheij made a small, but awesome, mistake. He went on eBay one evening and bid on a bunch of bulk LEGO brick auctions, then went to sleep. Upon waking, he discovered that he was the high bidder on many, and was now the proud owner of two tons of LEGO bricks. (This is about 4400 pounds.) He wrote, "[L]esson 1: if you win almost all bids you are bidding too high."

Mattheij had noticed that bulk, unsorted bricks sell for something like €10/kilogram, whereas sets are roughly €40/kg and rare parts go for up to €100/kg. Much of the value of the bricks is in their sorting. If he could reduce the entropy of these bins of unsorted bricks, he could make a tidy profit. While many people do this work by hand, the problem is enormous—just the kind of challenge for a computer. Mattheij writes:

There are 38000+ shapes and there are 100+ possible shades of color (you can roughly tell how old someone is by asking them what lego colors they remember from their youth).

In the following months, Mattheij built a proof-of-concept sorting system using, of course, LEGO. He broke the problem down into a series of sub-problems (including "feeding LEGO reliably from a hopper is surprisingly hard," one of those facts of nature that will stymie even the best system design). After tinkering with the prototype at length, he expanded the system to a surprisingly complex system of conveyer belts (powered by a home treadmill), various pieces of cabinetry, and "copious quantities of crazy glue."

Here's a video showing the current system running at low speed:

The key part of the system was running the bricks past a camera paired with a computer running a neural net-based image classifier. That allows the computer (when sufficiently trained on brick images) to recognize bricks and thus categorize them by color, shape, or other parameters. Remember that as bricks pass by, they can be in any orientation, can be dirty, can even be stuck to other pieces. So having a flexible software system is key to recognizing—in a fraction of a second—what a given brick is, in order to sort it out. When a match is found, a jet of compressed air pops the piece off the conveyer belt and into a waiting bin.

After much experimentation, Mattheij rewrote the software (several times in fact) to accomplish a variety of basic tasks. At its core, the system takes images from a webcam and feeds them to a neural network to do the classification. Of course, the neural net needs to be "trained" by showing it lots of images, and telling it what those images represent. Mattheij's breakthrough was allowing the machine to effectively train itself, with guidance: Running pieces through allows the system to take its own photos, make a guess, and build on that guess. As long as Mattheij corrects the incorrect guesses, he ends up with a decent (and self-reinforcing) corpus of training data. As the machine continues running, it can rack up more training, allowing it to recognize a broad variety of pieces on the fly.

Here's another video, focusing on how the pieces move on conveyer belts (running at slow speed so puny humans can follow). You can also see the air jets in action:

In an email interview, Mattheij told Mental Floss that the system currently sorts LEGO bricks into more than 50 categories. It can also be run in a color-sorting mode to bin the parts across 12 color groups. (Thus at present you'd likely do a two-pass sort on the bricks: once for shape, then a separate pass for color.) He continues to refine the system, with a focus on making its recognition abilities faster. At some point down the line, he plans to make the software portion open source. You're on your own as far as building conveyer belts, bins, and so forth.

Check out Mattheij's writeup in two parts for more information. It starts with an overview of the story, followed up with a deep dive on the software. He's also tweeting about the project (among other things). And if you look around a bit, you'll find bulk LEGO brick auctions online—it's definitely a thing!

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Health
One Bite From This Tick Can Make You Allergic to Meat
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We like to believe that there’s no such thing as a bad organism, that every creature must have its place in the world. But ticks are really making that difficult. As if Lyme disease wasn't bad enough, scientists say some ticks carry a pathogen that causes a sudden and dangerous allergy to meat. Yes, meat.

The Lone Star tick (Amblyomma americanum) mostly looks like your average tick, with a tiny head and a big fat behind, except the adult female has a Texas-shaped spot on its back—thus the name.

Unlike other American ticks, the Lone Star feeds on humans at every stage of its life cycle. Even the larvae want our blood. You can’t get Lyme disease from the Lone Star tick, but you can get something even more mysterious: the inability to safely consume a bacon cheeseburger.

"The weird thing about [this reaction] is it can occur within three to 10 or 12 hours, so patients have no idea what prompted their allergic reactions," allergist Ronald Saff, of the Florida State University College of Medicine, told Business Insider.

What prompted them was STARI, or southern tick-associated rash illness. People with STARI may develop a circular rash like the one commonly seen in Lyme disease. They may feel achy, fatigued, and fevered. And their next meal could make them very, very sick.

Saff now sees at least one patient per week with STARI and a sensitivity to galactose-alpha-1, 3-galactose—more commonly known as alpha-gal—a sugar molecule found in mammal tissue like pork, beef, and lamb. Several hours after eating, patients’ immune systems overreact to alpha-gal, with symptoms ranging from an itchy rash to throat swelling.

Even worse, the more times a person is bitten, the more likely it becomes that they will develop this dangerous allergy.

The tick’s range currently covers the southern, eastern, and south-central U.S., but even that is changing. "We expect with warming temperatures, the tick is going to slowly make its way northward and westward and cause more problems than they're already causing," Saff said. We've already seen that occur with the deer ticks that cause Lyme disease, and 2017 is projected to be an especially bad year.

There’s so much we don’t understand about alpha-gal sensitivity. Scientists don’t know why it happens, how to treat it, or if it's permanent. All they can do is advise us to be vigilant and follow basic tick-avoidance practices.

[h/t Business Insider]

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