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.)

Cut a notch in a stick and label the two parts p and q. Then draw the stick around the shore of a pond. The notch will describe a curve, and, remarkably, the area between the shore and this curve will be given by πpq.

“Two things immediately struck me as astonishing,” wrote British mathematician Mark Cooker in 1988. “First, the formula for the area is independent of the size of the given curve. Second, [the equation for the area] is the area of an ellipse of semi-axes p and q, but there are no ellipses in the theorem!”

For years Raymond Smullyan sought a “metaparadox,” a statement that is paradoxical if and only if it isn’t. He arrived at this:

Either this sentence is false, or (this sentence is paradoxical if and only if it isn’t).

He wrote, “I leave the proof to the reader.”