Forking Paths

British maze designer Randoll Coate produced this tribute to Jorge Luis Borges — a labyrinth of hedges shaped like an open book and spelling out the author’s name. (The original maze is in the writer’s native Argentina; Coate donated the copy above to Borges’ foundation in Venice.)

“Five years before Borges died, I had a dream in which I heard that Borges had just died,” the designer recalled. “And I thought to myself, I must make sure that Borges is not memorialized with one of those terrible statues — a depiction of angels or something. He has to be honored with something truly Borgesian, in other words, a labyrinth. That’s when I began to design it and think about it and dream up a shape for it — an extraordinary labyrinth for a man with an extraordinary mind.”

(From Francesca Tatarella, Labyrinths & Mazes, 2016.)

11/29/2017 UPDATE: Coate’s design is more sophisticated than I’d realized — from reader Daniël Hoek:

“The maze also contains a tiger, a walking stick, a question mark, the initials MK of his wife, and two hourglasses that spell the number of years Borges lived (’86’). After some effort I think I found all of those (the tiger is very cool once you find it –– you need to rotate the plan as in the attached image)”:

“PS. Another interesting tidbit: the maze in ‘The Garden of Forking Paths’ has the feature that you can make it through by going left at every turn. Starting at the top entrance and discounting any forced turns, that is also true of this maze, although that is a boring route that takes you around the maze and not through the center.”

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


On June 30, 1999, the body of 41-year-old Ricky McCormick was discovered near a cornfield in West Alton, Missouri. He’d last been seen alive five days earlier; now he was 15 miles from home though he owned no car. In his pockets were two handwritten notes (click to enlarge):

In the ensuing 18 years both the FBI’s Cryptanalysis and Racketeering Records Unit and the American Cryptogram Association have failed to find any meaning in these messages. In 2011 the FBI appealed to the public for their insights: If you have any you can contact them via this page.

“We are really good at what we do,” said CRRU chief Dan Olson, “but we could use some help with this one.”

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.”

Blades of Glory

I don’t know why this never caught on — in 1959 Klara Karwowska invented little windshield wipers for eyeglasses:

The present invention is directed to a wiper means for maintaining the lenses clean or clear of steam, rain, snow, or other foreign matter, and the wiper mechanism of the present invention includes a source of electrical energy such as the battery which may be secured to the frame in any suitable manner.

The battery would make them a little bulky, but that’s a small price to pay for clear vision. I could wear these in the shower!

In a Word

n. cessation of labor

v. to extend hospitality to

adj. disposed to follow a leader

adj. brought back to one’s senses

I spent an evening at the house of the president of Harvard University. The party was waited on at tea by a domestic of the president’s, who is also Major of the Horse. On cavalry days, when guests are invited to dine with the regiment, the major, in his regimentals, takes the head of the table, and has the president on his right hand. He plays the host as freely as if no other relation existed between them. The toasts being all transacted, he goes home, doffs his regimentals, and waits on the president’s guests at tea.

— Harriet Martineau, Society in America, 1837


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