Just a charming little anecdote: When German chemist Adolf von Baeyer achieved a long-sought result, he tipped his hat to it:

Eventually, however, even Baeyer was supersaturated with these hydrogenations, and the sorely tried assistants hailed with deep relief the transference of his interest to succinylsuccinic ester and diketocyclohexane. By means of a dodge (‘Kunstgriff’) of which Baeyer was very proud (treatment with sodium amalgam in presence of sodium bicarbonate), the diketone was reduced to quinitol. At the first glimpse of the crystals of the new substance Baeyer ceremoniously raised his hat!

It must be explained here that the Master’s famous greenish-black hat plays the part of a perpetual epithet in Prof. Rupe’s narrative. As the celebrated sword-pommel to Paracelsus, so this romantic hard-hitter or ‘alte Melone’ to Baeyer: the former was said to contain the vital mercury of the mediaeval philosophers; the latter certainly enshrined one of the keenest chemical intellects of the modern world. … Baeyer’s head was normally covered. Only in moments of unusual excitement or elation did the Chef remove his hat: apart from such occasions his shiny pate remained in permanent eclipse.

(From his colleague John Read’s 1947 book Humour and Humanism in Chemistry.)

Suppose we have a group of n men and n women. Each of the women can find some subset of the men whom she would be happy to marry. And each of the men would be happy with any woman who will have him. Is it always possible to pair everyone off into happy marriages?

Clearly this won’t work if, for example, two of the women have their hearts set on the same man and won’t be happy with anyone else. In general, for any subset of the women, we need to be sure that they can reconcile their preferences so that each of them finds a mate.

Surprisingly, though, that’s all that’s required. So long as every subset of women can collectively express interest in a group of men at least as numerous as their own, it will always be possible to marry off the whole group into happy couples.

The theorem was proved by English mathematician Philip Hall in 1935. Another application of the same principle: Shuffle an ordinary deck of 52 playing cards and deal it into 13 piles of 4 cards each. Now it’s always possible to assemble a run of 13 cards, ace through king, by drawing one card from each pile.

As early as the 1st century B.C., the Chinese text Zhou Bi Suan Jing reflected the reasoning of the Pythagorean theorem, showing how to find the hypotenuse of the 3-4-5 triangle. Arrange four 3×4 rectangles around a unit square, as shown, producing a 7×7 square. The diagonals of the four rectangles produce a tilted square. Now, the area of the 7×7 square is 49, and the area of one right triangle with legs 3 and 4 is 6. So the area of the tilted square is 49 – (4 × 6), or 25. This shows that the hypotenuse of each of the right triangles is 5.

In Mathematics and the Aesthetic (2007), Nathalie Sinclair writes, “The Chinese diagram … is the same as one given by the twelfth-century Indian scholar Bhaskara, whose one-word injunction Behold! recorded his sense of awe.”