A stunning geometric alphamagic square by Lee Sallows. The 3 × 3 grid is a familiar magic square in which each number is spelled out: The first cell contains the number 25, the second 2, and so on. Interpreted in this way, each row, column, and long diagonal sums to 45.

But there’s more: The English name of the number in each cell has been arranged onto a distinctive tile, such that the three tiles in any row, column, or long diagonal can be combined to form the same 21-cell figure, as shown. (Shapes with dotted outlines have been turned over.)

And yet more: Count the number of letters in each of the number names (or, equivalently, count the number of cells that make up each tile). So, for example, TWENTY-FIVE has 10 letters, so replace the TWENTYFIVE tile with the number 10. Similarly, replace TWO with 3, EIGHTEEN with 8, and so on. This produces another magic square:

10  3  8
 5  7  9
 6 11  4

Each row, column, and long diagonal totals 21.

I’m just sharing this because I think it’s pretty — it’s the smallest arrangement of identical non-crossing matchsticks that one can make on a tabletop in which each match-end touches three others.

Presented by German mathematician Heiko Harborth in 1986, it’s known as the Harborth graph.

A perplexing story from logician Raymond Smullyan:

Oh, one other thing. I must tell you of a certain great Sage in the East who was reputed to be the wisest man in the world. A philosopher heard about him and was anxious to meet him. It took him fifteen years to find him, but when he finally did, he asked him: ‘What is the best question that can be asked, and what is the best answer that can be given?’ The great Sage replied: ‘The best question that can be asked is the question you have asked, and the best answer that can be given is the answer I am now giving.’

It’s at the very end of his last book, A Mixed Bag, from 2016.

A normal die is painted so that it has four green faces and two red. Then it’s shaken in a cup and thrown repeatedly onto a table. You’re invited to guess which of these three sequences results. If you guess wrong you lose $10; and if you guess right you win $30.

1. RGRRR
2. GRGRRR
3. GRRRRR

Most people express the preferences 2, 1, 3, in that order. Red is less likely than green, but it predominates in all three sequences, so many subjects explain that sequence 2 is more “balanced,” and therefore more probable. In fact 65 percent of all subjects (excluding expert statisticians and people whose business is probability) show a strong propensity to vote for sequence 2, even when it’s pointed out explicitly that sequence 1 is just sequence 2 minus the first throw — so sequence 2 cannot be more likely!

“The longer the sequence, the less probable it is, independently of its being ‘balanced’ or ‘unbalanced,'” writes Massimo Piattelli-Palmarini in Inevitable Illusions. “This shows how resistant certain cognitive illusions are. Many other more complex examples have been advanced, and these show that even professional statisticians are sometimes subject to the same illusion.”

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.)

The Paradox of the Court is a logic problem from ancient Greece. Protogoras took on a pupil, Euathlus, on the understanding that Euathlus would pay him after he won his first court case. After Protogoras taught him the law, Euathlus decided not to practice, and Protogoras sued him for the amount owed.

Protagoras argued that if he won this lawsuit, he’d be paid the money he was owed, and if Euathlus won the suit, then he’d have won his first case and would owe Protagoras the money anyway under the terms of their contract. So he ought to be paid either way.

Euathlus argued that if he won the suit then by the court’s decision he owed nothing, and if he lost the suit then he still would not have won his first case, and thus owed Protagoras nothing under the contract.

One lawyer suggested that the court should decide in favor of the student and declare that he doesn’t have to pay for his education. Then Protagoras should sue him a second time — since then, incontrovertibly, the student will have won his first case!