Squeeze six circles into a larger circle so that each is tangent to its two neighbors. Now the three lines drawn through opposite points of tangency will pass through the same point.

Remarkably, this wasn’t discovered until 1974.

If a quadrilateral circumscribes a circle, then the sums of its opposite sides are equal.

Above, a + c = b + d.

Two travelers are transporting identical antiques. Unfortunately, the airline smashes both of them. The airline manager proposes that each traveler write down the cost of his antique, any value from $2 to $100. If both write the same number, the airline will pay this amount to both travelers. If they write different numbers, the airline will assume that the lower number is the accurate price; the low bidder will receive this amount plus $2, and the high bidder will receive this amount minus $2. If they can’t confer, what strategy should the travelers take in deciding how to bid?

At first Traveler A might like to bid $100, the maximum allowed. If his opponent does the same then they’ll both net $100. But A can do better than this: If B bids $100 and A bids $99 then A will come away with $101.

Unfortunately if B is rational then he’ll have the same insight and also bid $99. So A had better undercut him again and bid $98.

This chain leads all the way down to $2. If both travelers are perfectly rational then they’ll both bid (and make) $2, the minimum price.

But this seems very unlikely to happen in actual practice — in real life both travelers would likely make high bids and get high (though perhaps unequal) payoffs.

“All intuition seems to militate against all formal reasoning in the traveler’s dilemma,” wrote economist Kaushik Basu in propounding the problem in 1994. “There is something very rational about rejecting (2, 2) and expecting your opponent to do the same. … The aim is to explain why, despite rationality being common knowledge, players would reject (2, 2), as intuitively seems to be the case.”

A Japanese geometry theorem from the Edo period: If the blue circles are equal, the green circles will be equal too.

This can be extended: Circles spanning three of these triangles will also be equal, and so on.

Jump into the sea and look up. The surface above you is dark except for a bright circle that follows you around like a portable skylight. This is Snell’s window: Because light is refracted as it enters the water, the 180-degree world above you is compressed into a tight 97 degrees.

Physicist Robert W. Wood was thinking of this effect when he created a new wide-angle lens in 1906. Fittingly, he called it the fisheye.

In 1990, Spanish philosopher Jon Perez Laraudogoitia submitted an article to Mind entitled “This Article Should Not Be Rejected by Mind.” In it, he argued:

1. If statement 1 in this argument is trivially true, then this article should be accepted.
2. If statement 1 were false, then its antecedent (“statement 1 in this argument is trivially true”) would be true, which means that statement 1 itself would be true, a contradiction. So statement 1 must be true.
3. But that seems wrong, since Mind is a serious journal and shouldn’t publish trivial truths.
4. That means statement 1 must be either false or a non-trivial truth. We know it can’t be false (#2), so it must be a non-trivial truth, and its antecedent (“statement 1 in this argument is trivially true”) is false.
5. What then is the truth value of its consequent, “this article should be accepted”? If this were false then Mind shouldn’t publish the article; that can’t be right, since the article consists of a non-trivial truth and its justification.
6. So the consequent must be true, and Mind should publish the article.

They published it. “This is, I believe, the first article in the whole history of philosophy the content of which is concerned exclusively with its own self, or, in other words, which is totally self-referential,” Laraudogoitia wrote. “The reason why it is published is because in it there is a proof that it should not be rejected and that is all.”