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
Diagnosed with autism at 3, Stephen Wiltshire quickly became fascinated with drawing London buildings, and by age 8 he was sketching Salisbury Cathedral for former Prime Minister Edward Heath. Known as “the living camera,” he can draw an accurate, detailed picture of a subject after seeing it once — including subjects as complex as major cities. He’s completed enormous canvases of Tokyo, Rome, Hong Kong, Frankfurt, Madrid, Dubai, Jerusalem, and London, drawing each from memory after a helicopter ride.
Below is the full time-lapse of his portrait of Singapore, drawn from memory over five days in 2014. “That he has a gift makes no sense at all to Stephen,” his sister Annette told the New York Times. “He knows that he draws very well, but he picks that up from other people — he sees the warmth on their faces, they tell him how much they like his work, and that makes him very happy. He loves the attention.”
Visitors to Arizona’s Petrified Forest National Park sometimes can’t resist making off with a souvenir or two. Those who do sometimes return the stolen pieces with a “conscience letter” describing the misfortunes that have befallen them. Trinity Christian College art professor Ryan Thompson went through the 1200 pages in the park’s archives and collected the best of them for a 2014 book, Bad Luck, Hot Rocks. Some examples:
“Here are your rocks. Nothing but bad trouble.”
“Please put this back so my husband can get well. I tried to keep him from taking it.”
“Found this in my room. You can have it back. It’s bad luck. I got busted the other night.”
“I am sorry I took this. I am only 5 years old and made a bad mistake.”
And simply “You were right!”
There’s more at the website. Strangely, the same thing happens at Hawaii Volcanoes National Park, where an actual fire goddess will punish you for stealing lava rocks. And see The Conscience Fund.
“The difficulty of ruling over a diverse nation,” a 1578 engraving by Antwerp-based artist Pieter van der Borcht the Elder.
James A. Garfield wrote, “All free governments are managed by the combined wisdom and folly of the people.”
From the Public Domain Review.
In 1951, Arthur B. Brown of Queens College noted that the number 3 can be expressed as the sum of one or more positive integers in four ways (taking the order of terms into account):
3
1 + 2
2 + 1
1 + 1 + 1
As it turns out, any positive integer n can be so expressed in 2n – 1 ways. Brown asked, how can this be proved?
William Moser of the University of Toronto offered this insightful solution:
Imagine the digit 1 written n times in a row. For example, if n = 4:
1 1 1 1
This is a picket fence, with n pickets and n – 1 spaces between them. At each space we can choose either to insert a plus sign or leave it blank. So that gives us n – 1 tasks to perform (i.e., making this choice for each space) and two options for each choice. Thus the total number of expressions for n as a sum is 2n – 1, or, in the case of n = 4, eight:
1 1 1 1 = 4
1 + 1 1 1 = 1 + 3
1 1 + 1 1 = 2 + 2
1 1 1 + 1 = 3 + 1
1 + 1 + 1 1 = 1 + 1 + 2
1 + 1 1 + 1 = 1 + 2 + 1
1 1 + 1 + 1 = 2 + 1 + 1
1 + 1 + 1 + 1 = 1 + 1 + 1 + 1
(Pi Mu Epsilon Journal 1:5 [November 1951], 186.)
