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

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.
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.


15 Pi Day Gifts for the Math Fan in Your Life

March 14, the mathematic high holiday known as Pi Day, is right around the corner. To celebrate everyone's favorite irrational number, we've rounded up some gifts to help the math aficionados in your life—the ones who know that pi is the ratio of the circumference of a circle to its diameter—observe Pi Day in proper fashion.

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1. PI PIE PAN; $25

If Pi Day passed and you didn't eat a pi pie, did Pi Day even happen? This specially shaped baking pan makes the equivalent volume of a 9-inch round pan, but obviously has more surface area than a standard pan. Pi puns and extra crust? Sounds like a win-win dessert.

Find It: Amazon


Pair that pi pie with a set of these special plates decorated with a formula that spells out "imaginary unit eight summation pi"—or, essentially, "I ate some pie." Yes please!

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Inspire a love of irrational numbers in the young mathematician-to-be in your life with this adorable cotton onesie, available in five colors for 6-24 month olds.

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We do our best thinking in the shower, and this machine-washable shower curtain is sure to inspire a stumped mathematician to finally figure out x once and for all.

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Consider this equation: Math puns + affordability = this hilarious gift tee.

Find It: $6 Dollar Shirts


You'll be toasting to a gift well done after they open this set of four pint glasses measuring out the number of ounces in Pythagoras's constant, the Golden Ratio, Euler's Number, and of course pi.

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It's never too early to get your budding mathematician hooked on STEM! This quantum physics intro is meant for 1–3 year olds, but it's a good refresher for adults to brush up on their knowledge too.

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This "classroom pack" of temporary tattoos means that when you and 44 of your closest pi pals practice memorizing pi's numerous digits, you never have to leave home without your cheat sheet.

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Definitely know your audience before gifting this head-scratcher of a clock. For some, the regular mental exercise to figure out the time would be a welcomed brain-teaser. For others, it could be a frustrating distraction. But, we think its namesake—it should be relatively easy to figure out which Albert it's referencing—would be a fan.

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Spend your savasana meditating on the wonders of math in these equation-covered leggings, which come in sizes XS-4X.

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Ancient calculators make great toys when it comes to this colorful bead toy aimed at kids 2 and up. But once the young ones hit grade school, this specially marked abacus will help them visualize arithmetic while still seeing the equations listed out.

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This coloring book takes nature's best mathematical patterns and turns them into a soothing adult coloring book. Take a break from studying math's interconnected worlds, and just connect pencil to paper for a bit.

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Cookies are certainly easier to bake in bulk than pies. And if our math checks out, that means they will probably last a little longer too …

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This hands-on math game makes learning arithmetic engaging and entertaining, and can help kids 3–6 years old recognize units and solve basic additions and subtractions. These wooden letters come with three free apps that you pair with any iPad and most Samsung and Nexus tablets.

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You saw the movie—now delve even deeper into the true stories of Katherine Johnson, Dorothy Vaughan, Mary Jackson, and the other African-American women who worked at NASA as "human computers" during the Space Race. Margot Lee Shetterly's best-seller reveals just how much ground-breaking work these brilliant mathematicians truly did, even while dealing with both gender discrimination and the Jim Crow era. And if you haven't seen the movie, stream it on HBO or purchase it here.

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See If You Can Solve This Tricky Coin-Flipping Riddle

Make sure your head is in working order before trying to solve this riddle from TED-Ed, because it's a stumper.

Here's the scenario: You're an explorer who's just stumbled upon a trove of valuable coins in a remote dungeon. Each coin has a gold side and a silver side, each with an identical scorpion seal. The wizard who guards the coins agrees to let you have them, but he won't let you leave the room unless you separate the hoard into two piles with an equal number of coins with the silver side facing up in each. You've just counted the total number of silver-side-up coins—20—when the lights go out. In the dark, you have no way of knowing which half of a coin is silver and which half is gold. How do you divide the pile without looking at it?

As TED-Ed explains, the task is fairly easy to complete, no psychic powers required. All you need to do is remove any 20 coins from the pile at random and flip them over. No matter what combination of coins you choose, you will suddenly have a number of silver-side-up coins that's equal to whatever is left in the pile. If every coin you pulled was originally gold-side-up, flipping them would give you 20 more silver-side-up coins. If you chose 13 gold-side-up coins and seven of the silver-side coins, you'd be left with 13 silver coins in the first pile and 13 silver ones in your new stack after flipping it over.

The solution is simple, but the algebra behind it may take a little more effort to comprehend. For the full explanation and a bonus riddle, check out the video from TED-Ed below.

[h/t TED-Ed]


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