Mathematics

Von Osten firmly believed that humanity had greatly underestimated the reasoning skills and intelligence of animals. To test his hypothesis, he took it upon himself to tutor a cat, a horse, and a bear in the ways of mathematics. The cat was indifferent to his efforts, and the bear seemed outright hostile, but the arab stallion named Hans showed some real promise. With further tutelage, Hans the horse learned to use his hoof to tap out numbers written on a blackboard. Much to Von Osten's delight, jotting a "3" on the blackboard would prompt a tap-tap-tap from his pupil, a feat which Hans could repeat for any number under ten.

Encouraged by this success, Von Osten pressed his student further. The scientist drew out some basic arithmetic problems on his chalkboard, and attempted to train the horse in the symbols' meanings. Hans had no problem keeping up with the curriculum, and soon he was providing the correct responses to a variety of math problems including basic square roots and fractions. Hans was proving to be a clever horse indeed.

Consider the following example: Assuming for a moment that birthdays are evenly distributed throughout the year, if you're sitting in a room with forty people in it, what are the chances that two of those people have the same birthday? For simplicity's sake, we'll ignore leap years. A reasonable, intelligent person might point out that the odds don't reach 100% until there are 366 people in the room (the number of days in a year + 1)... and forty is about 11% of 366... so such a person might conclude that the odds of two people in forty sharing a birthday are about 11%. In reality, due to Math's convoluted reasoning, the odds are about 90%. This phenomenon is known as the Birthday Paradox.

Pierre de Fermat, a famous number theorist of the 17th century, rarely published his work – instead, he would often write comments in the margins of books. In one margin Fermat proposed that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2. However, rather than providing a proof, he only offered this taunting sentence: “I have discovered a truly remarkable proof which this margin is too small to contain.”

The proof for this simple conjecture was not solved for over 350 years and through the centuries became one of math's greatest puzzles.

If you ever plan to cheat on your taxes, here's something to consider (besides prison): Make sure that most of the numbers you fabricate start with the digit 1 (one). The second-most common leading digit should be 2, then 3, continuing on that pattern to leave 9 as the least common leading digit. This distribution is called Benford's Law, and it's a lot more straightforward than tax law... though why it exists is nearly as mysterious.

In a highly variable set of numbers such as those found in taxes, one would think that the leading digits would all be equally common. One would expect to find roughly the same amount of numbers starting with a 1 as, say, an 8. In a set of totally random numbers such as the lottery, that is exactly what one would discover; but when it comes to non-random real-life numbers, unless the data set is too constrained, a lot more numbers start with a one than any other digit. This can be useful in many ways.

On the show, once the contestant made their choice, Monty Hall (the host) opened one of the other doors, revealing one of the worthless prizes. He would then open the contestant's chosen door to reveal whether they picked correctly. The Monty Hall problem asks, what if the contestant were allowed to change her door choice after she saw the worthless prize? Would it be to her advantage to switch doors? In other words, if the contestant guesses that the new car lay behind door #1, and Monty opened door #2 to reveal a goat, is the new car more likely to be behind door #1, or door #3?

At this point, the imperfect wad of meat called the "brain" fires up it neurons, and usually informs its owner that revealing the contents of one of the other doors simply changed the contestant's odds from one-in-three to fifty-fifty. But that isn't the case. It has been mathematically proven that if the contestant were allowed to switch her door to #3 after seeing the goat behind #2, she'd be twice as likely to win.