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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.
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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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15 Riveting Facts About Alan Turing
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More than six decades after his death, Alan Turing’s life remains a point of fascination, even for people who have no interest in his groundbreaking work in computer science. He has been the subject of a play and an opera, and referenced in multiple novels and numerous musical albums. The Benedict Cumberbatch film about his life, The Imitation Game, received eight Oscar nominations. But just who was he in real life? Here are 15 facts you should know about Alan Turing.

1. HE’S THE FATHER OF MODERN COMPUTER SCIENCE.

Turing essentially pioneered the idea of computer memory. In 1936, Turing published a seminal paper called “On Computable Numbers” [PDF], which The Washington Post has called “the founding document of the computer age.” In the philosophical article, he hypothesized that one day, we could build machines that could compute any problem that a human could, using 0s and 1s. Turing proposed single-task machines called Turing machines that would be capable of solving just one type of math problem, but a “universal computer” would be able to tackle any kind of problem thrown at it by storing instructional code in the computer’s memory. Turing’s ideas about memory storage and using a single machine to carry out all tasks laid the foundation for what would become the digital computer.

In 1945, while working for the UK’s National Physical Laboratory, he came up with the Automatic Computing Machine, the first digital computer with stored programs. Previous computers didn’t have electric memory storage, and had to be manually rewired to switch between different programs.

2. HE PLAYED A HUGE ROLE IN WINNING WORLD WAR II.

Turing began working at Bletchley Park, Britain’s secret headquarters for its codebreakers during World War II, in 1939. By one estimate, his work there may have cut the war short by up to two years. He’s credited with saving millions of lives.

Turing immediately got to work designing a codebreaking machine called the Bombe (an update of a previous Polish machine) with the help of his colleague Gordon Welchman. The Bombe shortened the steps required in decoding, and 200 of them were built for British use over the course of the war. They allowed codebreakers to decipher up to 4000 messages a day.

His greatest achievement was cracking the Enigma, a mechanical device used by the German army to encode secure messages. It proved nearly impossible to decrypt without the correct cipher, which the German forces changed every day. Turing worked to decipher German naval communications at a point when German U-boats were sinking ships carrying vital supplies across the Atlantic between Allied nations. In 1941, Turing and his team managed to decode the German Enigma messages, helping to steer Allied ships away from the German submarine attacks. In 1942, he traveled to the U.S. to help the Americans with their own codebreaking work.

3. HE BROKE THE RULES TO WRITE TO CHURCHILL.

Early on, Bletchley Park’s operations were hampered by a lack of resources, but pleas for better staffing were ignored by government officials. So, Alan Turing and several other codebreakers at Bletchley Park went over their heads to write directly to Prime Minister Winston Churchill. One of the codebreakers from Bletchley Park delivered the letter by hand in October 1941.

“Our reason for writing to you direct is that for months we have done everything that we possibly can through the normal channels, and that we despair of any early improvement without your intervention,” they wrote to Churchill [PDF]. “No doubt in the long run these particular requirements will be met, but meanwhile still more precious months will have been wasted, and as our needs are continually expanding we see little hope of ever being adequately staffed.”

In response, Churchill immediately fired off a missive to his chief of staff: “Make sure they have all they want on extreme priority and report to me that this had been done.”

4. HE HAD SOME ODD HABITS.

Like many geniuses, Turing was not without his eccentricities. He wore a gas mask while riding his bike to combat his allergies. Instead of fixing his bike’s faulty chain, he learned exactly when to dismount to secure it in place before it slipped off. He was known around Bletchley Park for chaining his tea mug to a radiator to prevent it from being taken by other staffers.

5. HE RODE HIS BIKE 60 MILES TO GET TO THE FIRST DAY OF SCHOOL.

Though he was considered an average student, Turing was dedicated enough to his schooling that when a general strike prevented him from taking the train to his first day at his new elite boarding school, the 14-year-old rode his bike the 62 miles instead.

6. HE TRIED OUT FOR THE OLYMPICS.

Turing started running as a schoolboy and continued throughout his life, regularly running the 31 miles between Cambridge and Ely while he was a fellow at King’s College. During World War II, he occasionally ran the 40 miles between London and Bletchley Park for meetings.

He almost became an Olympic athlete, too. He came in fifth place at a qualifying marathon for the 1948 Olympics with a 2-hour, 46-minute finish (11 minutes slower than the 1948 Olympic marathon winner). However, a leg injury held back his athletic ambitions that year.

Afterward, he continued running for the Walton Athletic Club, though, and served as its vice president. ”I have such a stressful job that the only way I can get it out of my mind is by running hard,” he once told the club’s secretary. “It's the only way I can get some release."

7. HE WAS PROSECUTED FOR BEING GAY.

In 1952, Turing was arrested after reporting a burglary in his home. In the course of the investigation, the police discovered Turing’s relationship with another man, Arnold Murray. Homosexual relationships were illegal in the UK at the time, and he was charged with “gross indecency.” He pled guilty on the advice of his lawyer, and opted to undergo chemical castration instead of serving time in jail.

8. THE GOVERNMENT ONLY RECENTLY APOLOGIZED FOR HIS CONVICTION …

In 2009, UK Prime Minister Gordon Brown issued a public apology to Turing on behalf of the British government. “Alan and the many thousands of other gay men who were convicted as he was convicted under homophobic laws were treated terribly,” Brown said. "This recognition of Alan's status as one of Britain's most famous victims of homophobia is another step towards equality and long overdue." Acknowledging Britain’s debt to Turing for his vital contributions to the war effort, he announced, “on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better."

His conviction was not actually pardoned, though, until 2013, when he received a rare royal pardon from the Queen of England.

9. … AND NAMED A LAW AFTER HIM.

Turing was only one of the many men who suffered after being prosecuted for their homosexuality under 19th-century British indecency laws. Homosexuality was decriminalized in the UK in 1967, but the previous convictions were never overturned. Turing’s Law, which went into effect in 2017, posthumously pardoned men who had been convicted for having consensual gay sex before the repeal. According to one of the activists who campaigned for the mass pardons, around 15,000 of the 65,000 gay men convicted under the outdated law are still alive.

10. HE POISONED HIMSELF … MAYBE.

There is still a bit of mystery surrounding Turing’s death at the age of 41. Turing died of cyanide poisoning, in what is widely believed to have been a suicide. Turing’s life had been turned upside down by his arrest. He lost his job and his security clearance. By order of the court, he had to take hormones intended to “cure” his homosexuality, which caused him to grow breasts and made him impotent. But not everyone is convinced that he died by suicide.

In 2012, Jack Copeland, a Turing scholar, argued that the evidence used to declare Turing’s death a suicide in 1954 would not be sufficient to close the case today. The half-eaten apple by his bedside, thought to be the source of his poisoning, was never tested for cyanide. There was still a to-do list on his desk, and his friends told the coroner at the time that he had seemed in good spirits. Turing’s mother, in fact, maintained that he probably accidentally poisoned himself while experimenting with the chemical in his home laboratory. (He was known to taste chemicals while identifying them, and could be careless with safety precautions.)

That line of inquiry is far more tame than some others, including one author’s theory that he was murdered by the FBI to cover up information that would have been damaging to the U.S.

11. HIS FULL GENIUS WASN’T KNOWN IN HIS LIFETIME.

Alan Turing was a well-respected mathematician in his time, but his contemporaries didn’t know the full extent of his contributions to the world. Turing’s work breaking the Enigma machine remained classified long after his death, meaning that his contributions to the war effort and to mathematics were only partially known to the public during his lifetime. It wasn’t until the 1970s that his instrumental role in the Allies' World War II victory became public with the declassification of the Enigma story. The actual techniques Turing used to decrypt the messages weren’t declassified until 2013, when two of his papers from Bletchley Park were released to the British National Archives.

12. THE TURING TEST IS STILL USED TO MEASURE ARTIFICIAL INTELLIGENCE …

Can a machine fool a human into thinking they are chatting with another person? That’s the crux of the Turing test, an idea developed by Turing in 1950 regarding how to measure artificial intelligence. Turing argued in his paper “Computing Machinery and Intelligence” [PDF] that the idea of machines “thinking” is not a useful way to evaluate artificial intelligence. Instead, Turing suggests “the imitation game,” a way to assess how successfully a machine can imitate human behavior. The best measure of artificial intelligence, then, is whether or not a computer can convince a person that it is human.

13. … BUT SOME CONSIDER IT TO BE AN OUTDATED IDEA.

As technology has progressed, some feel the Turing test is no longer a useful way to measure artificial intelligence. It’s cool to think about computers being able to talk just like a person, but new technology is opening up avenues for computers to express intelligence in other, more useful ways. A robot’s intelligence isn’t necessarily defined by whether it can fake being human—self-driving cars or programs that can mimic sounds based on images might not pass the Turing test, but they certainly have intelligence.

14. HE CREATED THE FIRST COMPUTER CHESS PROGRAM.

Inspired by the chess champions he worked with at Bletchley Park, Alan Turing created an algorithm for an early version of computer chess—although at that time, there was no computer to try it out on. Created with paper and pencil, the Turochamp program was designed to think two moves ahead, picking out the best moves possible. In 2012, Russian chess grandmaster Garry Kasparov played against Turing’s algorithm, beating it in 16 moves. “I would compare it to an early caryou might laugh at them but it is still an incredible achievement," Kasparov said in a statement after the match-up.

15. THERE IS ALAN TURING MONOPOLY.

In 2012, Monopoly came out with an Alan Turing edition to celebrate the centennial of his birth. Turing had enjoyed playing Monopoly during his life, and the Turing-themed Monopoly edition was designed based on a hand-drawn board created in 1950 by his friend William Newman. Instead of hotels and houses, it featured huts and blocks inspired by Bletchley Park, and included never-before-published photos of Turing. (It’s hard to find, but there are still a few copies of the game on Amazon.)

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Why America Still Hasn’t Switched From Fahrenheit to Celsius
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If you grew up in America, you know that when the thermometer reads 32° F it’s time to bundle up, and if it’s 85° F outside you should break out a t-shirt. But say these temperatures to someone living in a different part of the world and you’ll likely be met with confusion. That’s because the United States joins Myanmar and Liberia as one of only three nations that don’t recognize the metric system.

In its new video, Vox explains why the U.S. is still measuring degrees in Fahrenheit long after the rest of the world decided to make the switch to metric. It wasn’t for the government’s lack of trying: In 1975, the country passed the Metric Conversion Act with the intention of selling the system to Americans. But while Canada, the UK, and Australia made adopting metric measurements mandatory, there was no such enforcement in the U.S. So, given the option to stick with what they know or teach themselves a whole new system, U.S. citizens chose the former.

To learn about the history of Fahrenheit and Celsius, and to see how the imperial system is more than just a nuisance for people visiting the U.S., check out Vox’s full report below.

[h/t Vox]

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