YouTube / Numberphile
YouTube / Numberphile

Should You Catch a Tossed Coin?

YouTube / Numberphile
YouTube / Numberphile

Here's the question: During a coin toss, is it more fair to catch the coin in mid-air, or to let it hit the ground? Meaning, will one of these methods result in any change to whether the coin comes up heads or tails? I was quite surprised to learn that apparently it does matter, even if it's just a little. (And I immediately thought of the Super Bowl, where the coin hits the ground and everybody stands around staring at it.)

In this video, randomness/coin-flipping expert Persi Diaconis talks us through the reality of biased coins and why they are biased. Then he gets to the meat of the question. I was genuinely surprised by his answer. Incidentally, Diaconis is also a former magician, but he's not talking about doing crazy hand-catching tricks; he assumes a typical human just catching a coin without trickery versus letting it hit the ground.

Somewhat related, here's a long examination of the recent Super Bowl coin tosses, and what was up with that 14-toss-winning streak by the NFC champions. Warning: lots of math in this one.

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Wikimedia Commons // Public Domain
15 Riveting Facts About Alan Turing
Wikimedia Commons // Public Domain
Wikimedia Commons // Public Domain

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, who was born on this day in 1912.

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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iStock
Honey Bees Can Understand the Concept of Zero
iStock
iStock

The concept of zero—less than one, nothing, nada—is deceptively complex. The first placeholder zero dates back to around 300 BCE, and the notion didn’t make its way to Western Europe until the 12th century. It takes children until preschool to wrap their brains around the concept. But scientists in Australia recently discovered a new animal capable of understanding zero: the honey bee. According to Vox, a new study finds that the insects can be taught the concept of nothing.

A few other animals can understand zero, according to current research. Dolphins, parrots, and monkeys can all understand the difference between something and nothing, but honey bees are the first insects proven to be able to do it.

The new study, published in the journal Science, finds that honey bees can rank quantities based on “greater than” and “less than,” and can understand that nothing is less than one.

Left: A photo of a bee choosing between images with black dots on them. Right: an illustration of a bee choosing the image with fewer dots
© Scarlett Howard & Aurore Avarguès-Weber

The researchers trained bees to identify images in the lab that showed the fewest number of elements (in this case, dots). If they chose the image with the fewest circles from a set, they received sweetened water, whereas if they chose another image, they received bitter quinine.

Once the insects got that concept down, the researchers introduced another challenge: The bees had to choose between a blank image and one with dots on it. More than 60 percent of the time, the insects were successfully able to extrapolate that if they needed to choose the fewest dots between an image with a few dots and an image with no dots at all, no dots was the correct answer. They could grasp the concept that nothing can still be a numerical quantity.

It’s not entirely surprising that bees are capable of such feats of intelligence. We already know that they can count, teach each other skills, communicate via the “waggle dance,” and think abstractly. This is just more evidence that bees are strikingly intelligent creatures, despite the fact that their insect brains look nothing like our own.

Considering how far apart bees and primates are on the evolutionary tree, and how different their brains are from ours—they have fewer than 1 million neurons, while we have about 86 billion—this finding raises a lot of new questions about the neural basis of understanding numbers, and will no doubt lead to further research on how the brain processes concepts like zero.

[h/t Vox]

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