Two by Two

kelly diagram

Cambridge mathematician Hallard T. Croft once asked whether it was possible to have a finite set of points in the plane with the property that the perpendicular bisector of any pair of them passes through at least two other points in the set.

In 1972 Leroy M. Kelly of Michigan State University offered the elegant solution above, a square with an equilateral triangle erected outward on each side (it also works if the triangles are erected inward).

“Croft is a great problemist,” Kelley said later. “He keeps putting out lists of problems and he keeps including that one. He’s trying to get the mathematical community to get a better example — one with more points in it. … Eight is the smallest number; and whether it’s the largest number is another question.”

So far as I know Croft’s question is still unanswered.

The Bean Machine

In 1894 Sir Francis Galton devised this simple machine to demonstrate the central limit theorem: Beads inserted at the top drop through successive rows of pegs, bouncing left or right as they hit each peg and landing finally in a row of bins at the bottom. Though the path of any given bead can’t be predicted, in the aggregate they form a bell curve. Delighted with this, Galton wrote:

Order in Apparent Chaos: I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the Law of Frequency of Error. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.

Square Deal

From Recreational Mathematics Magazine, 1961, a magic square of cards:

Each row, column, and main diagonal contains an ace, king, queen, and jack and all four suits. There are numerous other subsquares and symmetrical subsets of squares that have the same property, including the center 2 × 2 square and the four corner squares.

(Recreational Mathematics Magazine 34:5, 24-29, via Pi Mu Epsilon Journal, “Unusual Magic Squares,” 6:2 [Spring 1975], 54-55.)

The Chemists’ Drinking Song

In a 1963 essay, Isaac Asimov pointed out that paradimethylaminobenzaldehyde can be sung to the tune of “The Irish Washerwoman.” Inspired, John A. Carroll wrote this jig:

(chorus:) Paradimethylaminobenzaldehyde
Sodium citrate, ammonium cyanide
Phosphates and nitrates and chlorides galore
Just have one o’ these and you’ll never need more.

Got messed up last night on furfuryl alcohol
Followed it down with a gallon of propanol
Drank from mid-morning til late afternoon
Then spat on the floor and blew up the saloon.

(repeat chorus)

Powdered aluminum, nitrogen iodide
Slop it around and add some benzene
Then top off the punch with Fluorescein

(repeat chorus)

Whiskey, tequila and rum are too tame,
The stuff that I drink must explode into flame.
When I break wind it strips all the paint in the room,
And rattles the walls with an earthshaking boom.

(repeat chorus)

Go soak your head in a jar of formaldehyde
Scrub very hard, then rinse out your mane
In dichlorodiphenyltrichloroethane!

A Game Afoot

In “The Adventure of the Final Problem,” Sherlock Holmes flees London, pursued by his archenemy, James Moriarty. Both are headed to Dover, where Holmes hopes to escape to the continent, but there’s one intermediate stop available, at Canterbury. Holmes faces a choice: Should he get off at Canterbury or go on to Dover? If Moriarty finds him at either station he’ll kill him.

In their 1944 Theory of Games and Economic Behavior, mathematician John von Neumann and economist Oskar Morgenstern address this as a problem in game theory. They set up the following payoff matrix showing Moriarty’s calculations:

canterbury game

Von Neumann and Morgenstern conclude that “Moriarty should go to Dover with a probability of 60%, while Sherlock Holmes should stop at the intermediate station with a probability of 60% — the remaining 40% being left in each case for the other alternative.”

As it turns out, that’s exactly what happens in the story — Holmes and Watson get out at Canterbury and watch Moriarty’s train roar past toward Dover, “beating a blast of hot air into our faces.” “There are limits, you see, to our friend’s intelligence,” Holmes tells Watson. “It would have been a coup-de-maître had he deduced what I would deduce and acted accordingly.”

(It’s not quite that simple — in a footnote, von Neumann and Morgenstern point out that Holmes has excusably replaced the 60% probability with certainty in his calculations. In fact, they say, the odds favor Moriarty — “Sherlock Holmes is as good as 48% dead when his train pulls out from Victoria Station.”)


On Sept. 2, 1945, an American Navy squadron came ashore at Sagami Bay near Yokohama to demilitarize the Japanese midget submarines in the area. They found this notice on the door of a marine biological research station there, left by embryologist Katsuma Dan.

The Americans honored his wish: On the last of 1945 he was summoned by an officer of the U.S. First Cavalry and handed a document releasing the station back to the University of Tokyo.

The notice is on display at the Woods Hole Oceanographic Institution’s Marine Biological Laboratory (here’s the full story).

Hadwiger’s Conjecture

A triangle can be covered by three smaller copies of itself. A square requires four smaller copies. But in general four will do: Any bounded convex set in the plane can be covered with four smaller copies of itself (and in fact the fourth copy is needed only in the case of parallelograms, like the square).

Is this true in every dimension? In 1957 Swiss mathematician Hugo Hadwiger conjectured that every n-dimensional convex body can be covered by 2n smaller copies of itself. But this remains an unsolved problem.

(Interestingly, Russian mathematician Vladimir Boltyansky showed that this problem is equivalent to one of illumination: How many floodlights does it take to illuminate an opaque convex body from the exterior? The number of floodlights turns out to equal the number of smaller copies needed to cover the body.)

Spiral Tilings
Image: Wikimedia Commons

It’s easy to see that a plane can be tiled with squares or hexagons arranged in regular ranks, but in 1936 Heinz Voderberg showed that it can also be tiled in a spiral formation. Each tile in the figure above is the same nine-sided shape, but together they form two “arms” that bound one another. If both arms are extended infinitely, they’ll cover the whole plane.

In 1955 Michael Goldberg showed that spirals might be devised with any even number of arms, and in 2000 Daniel Stock and Brian Wichmann did the same for odd numbers, so it’s now possible to devise a shape that will tile the plane in a spiral with any specified number of arms.

(Daniel L. Stock and Brian A. Wichmann, “Odd Spiral Tilings,” Mathematics Magazine 73:5 [December 2000], 339-346.)

Do It Yourself

In the 19th century scientists were increasingly interested in comparing personality with brain anatomy, but they faced a problem: Lower-class brains could be acquired fairly easily from hospital morgues, but people with exceptional brains had the means to protect them from the dissecting knife after death.

The solution was the Society of Mutual Autopsy (Société d’autopsie mutuelle), founded in 1876 “for the purpose of furnishing to the investigations of medicists brains superior to those of the common people.” Anatomists bequeathed their brains to each other, and the results of each investigation were read out to the other members of the club. (An early forerunner was Georges Cuvier, whose brain was found to weigh 1830 grams and displayed a “truly prodigious number of convolutions.”)

Similar “brain clubs” sprang up in Munich, Paris, Stockholm, Philadelphia, Moscow, and Berlin before the practice began to die out around World War I. Until then, writes anthropologist Frances Larson in Severed, her 2014 history of severed heads, “Members could die happy in the knowledge that their own brain would become central to the utopian scientific project they had pursued so fervently in life.”

Early Delivery

In the 1940s British psychologist Robert H. Thouless set out to test the existence of life after death by publishing an enciphered message and then communicating the key to some living person after his own death. He published the following in the Proceedings of the Society for Psychical Research:


He wrote that “it uses one of the well-known methods of encipherment with a key-word which I hope to be able to remember in the after life. I have not communicated and shall not communicate this key-word to any other person while I am still in this world, and I destroyed all papers used in enciphering as soon as I had finished.” He hoped that his message would be unsolvable without supernatural aid because the message was relatively short and the cipher wasn’t simple. To prevent an erroneous decipherment, he revealed that his passage was “an extract from one of Shakespeare’s plays.” And he left the solution in a sealed envelope with the Society for Psychical Research, to be opened if this finally proved necessary.

He needn’t have worried — an unidentified “cipher expert” took up the cipher as a challenge and solved it in two weeks, long before Thouless’ death. It was the last two lines of this quotation from Macbeth:

Sleep that knits up the ravelled sleave of care
The death of each day’s life, sore labour’s bath
Balm of hurt minds, great nature’s second course,
Chief nourisher in life’s feast.

(It’s a Playfair cipher — a full solution is given in Craig Bauer’s excellent Unsolved!, 2017.)

Interestingly, Thouless published two other encrypted ciphers before his death in 1984, and only one has been solved. If you can communicate with the dead perhaps you can still solve it — it’s given on Klaus Schmeh’s blog.