Suppose you’re working on an algebraic expression that involves variables, addition, multiplication, and parentheses. You try repeatedly to expand it using the distributive law. How do you know that the expression won’t continue to expand forever?
For example, expanding
(x + y)(s(u + v) + t)
gives
x(s(u + v) + t) + y(s(u + v) + t),
which has more parentheses than the original expression.
“To believe entails no desire to know; everybody reads the Bible, but who reads Flavius Josephus?” — Arthur Koestler
To kill some time before a meeting of chess grandmasters, Burt Hochberg offered this anonymous puzzle from the 15th century. White must place four white rooks on the board, one at a time, giving check with each one. After each placement the black king can respond with any normal legal move. How can White plan his moves so that the fourth rook reliably gives checkmate?
There’s no trick, and in fact there are several solutions, but Hochberg says the grandmasters studied the position for several minutes before Paul Keres came up with an answer. What was it?
Working alone in his fields on June 8, 1948, Saskatchewan farmer Cecil George Harris accidentally put his tractor into reverse. It rolled backward, pinning his left leg under the rear wheel. His wife didn’t find him until 10:30 that night, and he died at the hospital.
Days later, surveying the scene of the accident, neighbors noticed that Harris had scratched an inscription into the tractor’s fender using his pocketknife:
In case I die in this mess, I leave all to the wife. Cecil Geo Harris.
The courts determined this to be a valid will. The fender was kept at the Kerrobert Courthouse until 1996; today it and the knife are displayed at the University of Saskatchewan law library.
06/08/2024 UPDATE: Geoff Ellwand and his colleagues published an impressively detailed account of the incident and its aftermath in the Saskatchewan Law Review in 2014. (Thanks, Tyler.)
This is said to have been the most popular problem presented in the American Mathematical Monthly. It was proposed by P.L. Chessin of Westinghouse in the April 1954 issue. Each of the digits in this long division problem has been replaced with an x — except for a single 8 in the quotient. Can you reconstruct the problem?
In this week’s episode of the Futility Closet podcast we’ll explore some more curiosities and unanswered questions from Greg’s research, including a pilot who saved Buckingham Palace, a ghost who confronted Arthur Conan Doyle, what Mark Twain learned from a palm reader, and a bedeviling superfluity of Norwegians.
We’ll also discover a language used only by women and puzzle over a gift that’s best given sparingly.
Here’s a surprise: A pendulum can be made stable in its inverted position if its support is oscillated rapidly up and down.
Even more improbably, in 1993 David Acheson and Tom Mullin showed that a double and even a triple pendulum can be made to stand up vertically like this if the pivot vibrates at the right frequency.
“The ‘trick’ really did work, and it worked, in fact, far better than we could ever have imagined,” Acheson wrote in 1089 and All That. “We were quite taken aback by just how stable the inverted state could be, and provided the pendulums were kept roughly aligned with one another, we could push them over by as much as 40 degrees or so and they would still gradually wobble back to the upward vertical.”
They published their results in Nature and later appeared on the BBC program Tomorrow’s World, but their demonstration didn’t impress everyone — afterward, an outraged caller upbraided the producers for “lowering their usually high standards” and “falling prey to two tricksters from Oxford.”
Here’s the double pendulum (thanks, Eccles):
From reader Magnus Ehinger: A mouse has apparently opened a restaurant and nut store in Malmö, Sweden, just outside the Kebab House at the intersection of Bergsgatan and Almbacksgatan. The restaurant is called Il Topolino (the Italian name for Mickey Mouse), and the nut store next door is Noix de Vie (“nuts of life”).
The restaurant offers a variety of cheese and crackers, according to the tiny menu posted outside, and the nut store offers pistachios, almonds, and hazelnuts. Also arranged outside are a tiny bicycle and posters for mouse-related films (including Night of the Were-Rat).
No one knows who’s behind this — a group called Anonymouse posted images on an Instagram account as this project took shape, and recently left an update reading “Without spoiling too much we can tell you that we’re working on a new scene, and in 2017 you’re going to be able to see plenty more.”
More information, and photos, are here. (That’s an English translation — here’s the article in the original Swedish. Thanks, Magnus.)
Dutch anatomist Frederik Ruysch had a curious sideline: He arranged fetal skeletons into allegorical dioramas on death and the transience of life. Set amid landscapes made of gallstones, kidney stones and preserved blood vessels, the skeletons are decorated with symbols of short life — one holds a mayfly, another weeps into a handkerchief made of brain meninges. Worms made of intestine wind through their rib cages. “Quotations and moral exhortations, emphasizing the brevity of life and the vanity of earthly riches, festooned the compositions,” notes Stephen Jay Gould in Finders, Keepers. “One fetal skeleton holding a string of pearls in its hand proclaims, ‘Why should I long for the things of this world?’ Another, playing a violin with a bow made of a dried artery, sings, ‘Ah fate, ah bitter fate.'”
Johannes Brandt, a Remonstrant teacher, wrote:
Oh, what are we? What remains of us when we are dead?
Behold, it is no living thing, but dry, bare bone instead.
Bladder stones you see in heaps, piled higher by the morrow:
Here one learns about life’s course through storms of pain and sorrow.
These wise lessons Ruysch presents with wit and erudition,
Amsterdam is fortunate to have this great physician.
One of the most famous cat-and-bird friendships on record was that between Caruso, a canary which belonged to President Coolidge, and Timmie, a black-and-white cat owned by Bascom Timmons, a Washington newspaperman. They became acquainted when Timmons took his cat to the White House, and Coolidge eventually sent the canary to Timmons’ home to live with the cat. After that they spent an hour or two every day together, the canary walking up and down the cat’s back or resting between his paws. According to Timmons, the canary fell over dead while singing to the cat.
— Roger Butterfield, “Cats,” LIFE, April 22, 1946
