Here’s something amazing — a machine made of fractions:

Start with the number 2 as your seed. Multiply it by each of the fractions above, in order, until you find one that produces an integer. (It’s 15/2.) Now adopt that integer (15) as the new seed, and multiply that by each of the fractions until you produce another integer. Keep this up, making a note whenever you produce a power of 2.

The first such power (4, or 2²) appears after 19 steps. Fifty steps later, 2³ turns up. Then 2⁵ appears about 200 steps further on. A pattern emerges: the exponents are 2, 3, 5 …

It turns out that “these fourteen fractions alone have it in them to produce an infinity of primes, even those that no one yet knows about,” writes Dominic Olivastro. “There is something enormously magical about it.” John Horton Conway devised the technique; it’s an instance of his Fractran computing algorithm.

I once had a friend who objected to assigning chores by lot on the grounds that random selection was biased in favour of lucky people. He claimed to be serious and went on to compare unlucky people with … groups he took to be victims of discrimination. Sincere or not, wherein lies the absurdity of my friend’s objection?

— Roy A. Sorensen, Blindspots, 1988

The thirsty but impecunious soul approaches the bar-tender with a request for brandy, or what not. He takes a sip, pronounces it detestable, and offers to change it for a glass of whiskey. The obliging bar-tender substitutes the whiskey. The customer drinks, smacks his lips, and prepares to depart. ‘Here,’ says the bar-tender, ‘you haven’t paid for your whiskey.’ ‘No,’ is the innocent response; ‘I gave you the brandy in exchange for it.’ ‘But you didn’t pay for the brandy.’ ‘But I didn’t drink it.’ And while the publican intellect is vainly struggling with the mathematical puzzle involved, the puzzler makes good his escape.

— William Shepard Walsh, Handy-Book of Literary Curiosities, 1892

A woman approached Bertrand Russell after a lecture.

“I’m so happy to find that you’re a solipsist,” she told him. “I’ve been one all my life, and I’m surprised there aren’t more of us.”

There is a form of boat-racing, occasionally used at regattas, which affords a somewhat curious illustration of certain mechanical principles. The only thing supplied to the crew is a coil of rope, and they have, without leaving the boat, to propel it from one point to another as rapidly as possible. The motion is given by tying one end of the rope to the after thwart, and giving the other end a series of violent jerks in a direction parallel to the keel. I am told that in still water a pace of two or three miles an hour can be thus attained.

The chief cause for this result seems to be that the friction between the boat and the water retards all relative motion, but it is not great enough to affect materially motion caused by a sufficiently big impulse. Hence the usual movements of the crew in the boat do not sensibly move the centre of gravity of themselves and the boat, but this does not apply to an impulsive movement, and if the crew in making a jerk move their centre of gravity towards the bow n times more rapidly than it returns after the jerk, then the boat is impelled forwards at least n times more than backwards: hence on the whole the motion is forwards.

— W.W. Rouse Ball, Mathematical Recreations and Essays, 1905

2 × 5 × 27 = 1 × 15 × 18
2 + 5 + 27 = 1 + 15 + 18

213 × 624 = 312 × 426
102 × 402 = 201 × 204
936 × 213 = 639 × 312

Pick any point inside an equilateral triangle and measure the distances to its sides.

The sum of those distances is the altitude of the triangle.