Discovered by Galileo:

A “proof without words” is here.

This is pretty: If you choose n > 1 equally spaced points on a unit circle and connect one of them to each of the others, the product of the lengths of these chords equals n.

(Andre P. Mazzoleni and Samuel Shan-Pu Shen, “The Product of Chord Lengths of a Circle,” Mathematics Magazine 68:1 [February 1995], 59-60.)

(Demonstration by Jay Warendorff.)

Suppose we divide a group of 100 women randomly into two groups, one of 95 and the other of 5 women. Then by flipping a fair coin we randomly assign the name “the Heads group” to one group and “the Tails group” to the other. Now philosopher John Leslie suggests that “[e]stimated probabilities can be observer-relative in a somewhat disconcerting way”:

All these persons — the women in the Heads group, those in the Tails group, and the external observer — are fully aware that there are two groups, and that each woman has a ninety-five per cent chance of having entered the larger. Yet the conclusions they ought to derive differ radically. The external observer ought to conclude that the probability is fifty per cent that the Heads group is the larger of the two. Any woman actually in [either the Heads or the Tails group], however, ought to judge the odds ninety-five to five that her group, identified as ‘the group I am in’, is the larger, regardless of whether she has been informed of its name.

Even without knowing her group’s name, a woman could still appreciate that the external observer estimated its chance of being the larger one as only fifty per cent — this being what his evidence led him to estimate in the cases of both groups. The paradox is that she herself would then have to say: ‘In view of my evidence of being in one of the groups, ninety-five per cent is what I estimate.’

Whether this is really a paradox is disputed — Oxford philosopher Nick Bostrom, in his 2002 book Anthropic Bias, points out that these are different judgments: The woman is considering the probability that she finds herself in the larger group, and the external observer is considering the assignment of labels to the groups. Bostrom describes a variant “where chances are observer relative in an interesting, but not paradoxical way” (page 129).

(John Leslie, “Observer-Relative Chances and the Doomsday Argument,” Inquiry 40:4 [1997], 427-436.)

{13, 40, 45}

The square of the sum of any two of these numbers minus the square of the third is a square:

(13 + 40)² – 45² = 28²
(13 + 45)² – 40² = 42²
(40 + 45)² – 13² = 84²

(From Edward Barbeau’s Power Play, 1997.)

A curious phenomenon noted in Martyn Cundy and A.P. Rollett’s Mathematical Models (1951):

Two rollers are mounted on perpendicular axles in different planes. An endless thread passes round them and connects them, both directly and with a crossover, as shown in the diagram. The instrument is somewhat capricious, but the following phenomena can be demonstrated with it.

(a) One roller is rotated continuously in one direction. The other starts in one direction, but if temporarily stopped with the finger continues in the opposite direction.

(b) One roller is rotated to and fro through a small angle. The other roller rotates continuously in the same direction.

“The apparatus shows that dynamical friction is less than statical, but a full explanation is complicated, if indeed it is possible, and certainly involves consideration of the elasticity of the connecting belt.”

In 2014, Princeton mathematician John Conway proposed a simple conjecture:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2² × 3 × 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(2² × 3 × 5) = 2235. Next, because 2235 = 3 × 5 × 149, it maps, under f, to 35149, and since 35149 is prime, we stop there forever.

The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → …, eventually getting to more than one hundred digits without reaching a prime!

Conway offered $1,000 for a counterexample, a number that doesn’t “climb to a prime.” The challenge stood for three years before James Davis, who calls himself “not a mathematician by any stretch,” found the pretty 13532385396179 = 13 × 53² × 3853 × 96179, a composite number that loops onto itself and thus never reaches a prime — a find worth $1,000.

It’s been established that a volume of fluid can be suspended stably above a layer of air by vibrating the whole system vertically. Now researchers in Paris have shown that lightweight objects can “float” on the underside of this suspended slab of liquid — thus inverting an entire seaside scene.

A YouTube commenter writes, “We’ve been sailing like that for years in Australia.”

(Benjamin Apffel et al., “Floating Under a Levitating Liquid,” Nature 585:7823 [2020], 48-52.)