Growing Pains

The Greek philosopher Democritus propounded this puzzle in the fourth century B.C.E.:

If a cone were cut by a plane parallel to the base, how must one conceive of the surfaces of the segments: as becoming equal or unequal? For being unequal, they make the cone irregular, taking many step-like indentations and roughnesses. But if they are equal, the segments will be equal and the cone will appear to have the property of the cylinder, being composed of equal, and not unequal, circles; which is most absurd.

If the cone’s cross section is increasing continuously, how can the two faces created by a cut fit together? It seems that one must be larger than the other, and yet at the same time it can’t be. How can we make sense of this?

Truth and Purity

In 2014 England’s University of Sheffield unveiled “the world’s first air-cleansing poem,” four stanzas by literature professor Simon Armitage that are printed on a 10-by-20-meter panel coated with particles of titanium dioxide that use sunlight and oxygen to clear the air of nitrogen oxide pollutants.

“This is a fun collaboration between science and the arts to highlight a very serious issue of poor air quality in our towns and cities,” said science professor Tony Ryan, who collaborated on the project. “This poem alone will eradicate the nitrogen oxide pollution created by about 20 cars every day.”

Armitage said, “Poetry often comes out with the intimate and the personal, so it’s strange to think of a piece in such an exposed place, written so large and so bold. I hope the spelling is right!”

Knife Fight

How can three people divide a cake so that none feels that another has a larger piece than his own? The Selfridge–Conway procedure, devised by mathematicians John Selfridge and John Horton Conway, will solve the problem in at most 5 cuts; it’s been called “one of the prettiest in the subject of cake cutting.”

Call the three participants Tom, Dick, and Harry. Tom begins by cutting the cake into three pieces that he regards as equal. Tom will be free of envy no matter how these are distributed, because he thinks they’re all the same. Now if Dick and Harry have different opinions as to which piece is largest, then everyone’s happy; we can divide the cake with no conflict.

But if both Dick and Harry both have their eyes on the same piece, then we have a problem — one of them is going to envy the other. The answer is to do some trimming: Dick trims the largest piece (in his eyes) until it matches the second-largest piece in size. Set the trimmings aside for the moment. (If Dick thinks the top two pieces are equal then no trimming is necessary.)

Now both Tom and Dick feel there’s more than one piece tied for biggest. So let Harry have his choice; this guarantees that he’ll be satisfied. This will leave behind at least one of Dick’s top two pieces, which he can have (if both are available then we insist he take the one he trimmed). And now Tom gets the remaining piece, which must be an untrimmed one, so he can have no objection.

What about the trimmings? Well, Tom got one of the untrimmed pieces, and he thought he made the inital cuts equitably, so he can have no objection if the trimmings (or any portion of them) go to the person who got the trimmed piece. Suppose that’s Dick. Have Harry divide the trimmings into three equal portions, and then have Dick choose first, Tom second, and Harry third. Dick is happy because he gets first choice, Tom can’t envy him for the reason just stated, and Harry cut the pieces to be equal, so he can’t feel envy either. Each of the three should be happy with his lot.

(Jack Robertson and William Webb, Cake-Cutting Algorithms, 1998.)


From Martin Gardner: Each of the two equal sides of an isosceles triangle is one unit long. How long must the third side be to maximize the triangle’s area? There’s an intuitive solution that doesn’t require calculus.

Click for Answer

Good Boy

Elisabeth Mann Borgese taught her dog to type. In her book The Language Barrier she explains that her English setter, Arli, developed a vocabulary of 60 words and 17 letters, though “He isn’t an especially bright dog.” “[Arli] could write under dictation short words, three-letter words, four-letter words, two-letter words: ‘good dog; go; bad.’ And he would type it out. There were more letters but I never got him to use more than 17.”

She began in October 1962 by training all four of her dogs to distinguish 18 designs printed on saucers; Arli showed the most promise, so she focused on him. By January 1963 he could count to 4 and distinguish CAT from DOG. Eventually she gave him a modified typewriter with enlarged keys, which she taught him to nose mechanically by rewarding him with hamburger. “No meaning at all was associated with the words,” she writes, though he did seem to associate meaning with words that excited him. “When asked, ‘Arli, where do you want to go?’ he will unfailingly write CAR, except that his excitement is such that the ‘dance’ around the word becomes a real ‘stammering’ on the typewriter. ACCACCAAARR he will write. GGOGO CAARR.”

(And it’s always tempting to discover meaning where there is none. Once while suffering intestinal problems after a long flight Arli ignored his work when she tried to get him to type GOOD DOG GET BONE, and then he stretched, yawned, and typed A BAD A BAD DOOG. This was probably just a familiar phrase that he’d chosen at random; Borgese estimated its likelihood at 1 in 12.)

Arli did earn at least one human fan — at one point Borgese showed his output to a “well-known critic of modern poetry,” who responded, “I think he has a definite affinity with the ‘concretist’ groups in Brazil, Scotland, and Germany [and an unnamed young American poet] who is also writing poetry of this type at present.”


In 1960, British researcher Donald Michie combined his loves of computation and biology to consider whether a machine might learn — whether by consulting its record of past experience it could perform tasks with progressively greater success.

To investigate this he designed a machine to play noughts and crosses (or tic-tac-toe). He called it the Machine Educable Noughts And Crosses Engine, which gives it the pleasingly intimidating acronym MENACE. MENACE consists of 304 matchboxes, each of which represents a board position. Each box contains a collection of beads representing available moves in that position, and after each game these collections are adjusted in light of the outcome (as described here). In this way the engine learns from its experience — over time it becomes less likely to play losing moves, and more likely to play winning (or drawing) ones, and it becomes a more successful player as a result.

University College London mathematician Matthew Scroggs describes the engine above, and he’s built an online version that you can try out for yourself — it really does get noticeably better as it plays.

Near and Far

Designed by Baroque architect Francesco Borromini in 1632, this gallery in Rome’s Palazzo Spada is a masterpiece of forced perspective — though it appears to be 37 meters long, in fact it’s only 8. The effect is produced by diminishing columns and a rising floor; the sculpture at the end, which Borromini contrived to appear life size, is only 60 centimeters high.
Image: Wikimedia Commons

That Settles That

The famous mathematician Stanislaw Ulam thought of the following paradox, which is now known as the Ulam Paradox: When President Richard Nixon was appointed to office, on the first day he met his cabinet he said to them: ‘None of you are yes-men, are you?’ And they all said, ‘NO!’

— Raymond Smullyan, A Mixed Bag, 2016


holmes circles

I say that conceit is just as natural a thing to human minds as a centre is to a circle. But little-minded people’s thoughts move in such small circles that five minutes’ conversation gives you an arc long enough to determine their whole curve. An arc in the movement of a large intellect does not sensibly differ from a straight line. Even if it have the third vowel [‘I’, the first-person pronoun] as its centre, it does not soon betray it. The highest thought, that is, is the most seemingly impersonal; it does not obviously imply any individual centre.

— Oliver Wendell Holmes Sr., The Autocrat of the Breakfast-Table, 1858


“Once I saw a chimpanzee gaze at a particularly beautiful sunset for a full 15 minutes, watching the changing colors until it became so dark that he had to retire to the forest without stopping to pick a pawpaw for supper.” — Adriaan Kortlandt