In 2001, USCD psychologist Vilayanur Ramachandran presented these shapes to American undergraduates and to Tamil speakers in India. He asked, “Which of these shapes is bouba and which is kiki?” Around 98% of the respondents assigned the name kiki to the spiky shape and bouba to the curvy one.

Psychologist Wolfgang Köhler had found a similar effect in 1929 using the words baluba and takete. “This result suggests that the human brain is somehow able to to extract abstract properties from the shapes and sounds,” Ramachandran wrote, “for example, the property of jaggedness embodied in both the pointy drawing and the harsh sound of kiki.”

When you’re a princess you have to kiss a lot of frogs. But you can see only one frog at a time, and once you reject a frog you can’t return to it. How can you know when to stop hoping for a prince and settle down with the frog you have?

“Surprisingly, … there is a method which enables us to select the best candidate with a probability of nearly 30% even if n is a large number,” writes Gabor Szekely in Paradoxes in Probability Theory and Mathematical Statistics (2001). “Let the first 37% (more precisely, 100/e%) of the candidates go and then select the first one better than any previous candidate (if none are better, select the last). In this case the chance of selecting the best is approximately 1/e, i.e. ≈37% however great n is.”

So if there are 100 frogs in the forest, reject the first 37 and then choose the first one that seems to beat all the others. There’s about a 37 percent chance that it’s the best one.

What is directly on the opposite side of the world from you? This map answers that question by superimposing each point on earth with its opposite. Some notable sisters:

• Beijing, China, is nearly opposite Buenos Aires, Argentina
• Madrid, Spain, is nearly opposite Wellington, New Zealand
• Bogotá, Colombia, is nearly opposite Jakarta, Indonesia
• Bangkok, Thailand, is nearly opposite Lima, Peru
• Quito, Ecuador, is nearly opposite Singapore
• Seoul, South Korea, is nearly opposite Montevideo, Uruguay
• Perth, Australia, is nearly opposite Bermuda
• Charmingly, Cherbourg, France, is opposite the Antipodes Islands south of New Zealand

W.V.O. Quine explains how an enterprising traveler can arrange to visit two precise antipodes: “Note to begin with that any route from New York to Los Angeles, if not excessively devious, is bound to intersect any route from Winnipeg to New Orleans. Now let someone travel from New York to Los Angeles, and also travel from roughly the antipodes of Winnipeg to roughly the antipodes of New Orleans. These two routes do not intersect — far from it; but one of them intersects a route that is antipodal to the other. So our traveler is assured of having touched a pair of mutually antipodal points precisely, though he will know only approximately where.”

In 2007, New Scientist announced that the best strategy in a game of rock paper scissors is to choose scissors.

Research has shown that rock is the most popular of the three moves. If your opponent expects you to choose it, he’ll choose paper in order to beat it — in which case scissors will win.

In 2005 a Japanese art collector asked Christie’s and Sotheby’s to play a match, saying the winner could sell his impressionist paintings. The 11-year-old daughter of a Christie’s director recommended scissors, saying, “Everybody expects you to choose rock.”

Sure enough, Sotheby’s chose paper, and Christie’s won the £10 million deal.

If there was a time when nothing existed, then there must have been a time before that — when even nothing did not exist. Suddenly, when nothing came into existence, could one really say whether it belonged to the category of existence or of non-existence?

— Chuang-Tzu

A hairy ball can’t be combed flat — it must always have a cowlick.

This result arose originally in algebraic topology, but it has intriguing applications elsewhere. For example, it can’t be windy everywhere at once on Earth’s surface — at any given moment, the horizontal wind speed somewhere must be zero.

In 1829 a correspondent to the Mechanic’s Magazine proposed this design for a “self-moving railway carriage.” Fill the car with passengers and cargo as shown and set it on two rails that undulate across the landscape:

In the descending sections (a, c, e) the two rails are parallel. In the ascending ones (b, d) they diverge so that the car, mounted on cones, will roll forward to settle more deeply between them, paradoxically “ascending” the slope. If the track circles the world the car will “assuredly continue to roll along in one undeviating course until time shall be no more.”

“How any one could ever imagine that such a contrivance would ever continue in motion for even a short time … must be a puzzle to every sane mechanic,” wrote John Phin in The Seven Follies of Science in 1911. But what does he know?

As a joke, Michael Collins submitted a travel voucher for his trip aboard Gemini 10. NASA reimbursed him $8 per day, a total of $24.

In his autobiography, Collins notes that he could instead have claimed 7 cents a mile, which would have yielded $80,000.

But one of the original Mercury astronauts had already tried this — and had received a bill for “a couple of million dollars” for the rocket he’d used.

Draw two circles of any size and bracket them with tangents, as shown.

The chords in green will always be equal.

Any group of six people must contain at least three mutual friends or three mutual strangers.

Represent the people with dots, and connect friends with blue lines and strangers with red. Will the completed diagram always contain a red or a blue triangle?

Because A has five relationships and we’re using two colors, at least three of A’s connections must be of the same color. Say they’re friends:

Already we’re perilously close to completing a triangle. We can avoid doing so only if B, C, and D are mutual strangers — in which case they themselves complete a triangle:

We can reverse the colors if B, C, and D are strangers to A, but then we’ll get the complementary result. The completed diagram must always contain at least one red or blue triangle.

I think this problem appeared originally in the William Lowell Putnam mathematics competition of 1953. Six is the smallest number that requires this result — a group of five people would form a pentagon in which the perimeter might be of one color and the internal connections of another.

(Update: In fact the more general version of this idea was adduced in 1930 by Cambridge mathematician F.P. Ramsey. It is very interesting.) (Thanks, Alex.)