The More the Airier?

In 1921 aeronautical engineer Giovanni Caproni designed a 100-seat transatlantic airliner with nine wings. With an empty weight of 14,000 kg, the Caproni Ca.60 Transaereo did tolerably well on its first test flight on Lake Maggiore, but it crashed on the second and never flew again. Caproni said, “So the fruit of years of work, an aircraft that was to form the basis of future aviation, all is lost in a moment. But one must not be shocked if one wants to progress. The path of progress is strewn with suffering.”

Nine wings isn’t even the record — that might belong to the “clever but somewhat dogmatic” Victorian engineer Horatio Phillips, who devised aircraft with up to 200 airfoils, basing them on a multi-vaned marine hydrofoil that he had designed. “But air and water do not behave similarly,” notes James Gilbert in The World’s Worst Aircraft (1976). “Air is compressible, while water, as you will know if you have ever belly-flopped into a swimming pool, hardly is. Multiple vanes lift well in water, poorly if at all in air.” Phillips spent £4,000 and gave up.

December 15, 2016 | Technology

Growth Potential

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.

	Select Click for Answer
Set all the variables equal to 2! “The point of the distributive law is that its application doesn’t change the value of the expression,” writes Dartmouth mathematician Peter Winkler. “The value of the initial expression limits the size of anything you can get from it by expansion.” Winkler received the puzzle from Dick Lipton of the College of Computing at Georgia Tech and included it in his “Seven Puzzles You Think You Must Not Have Heard Correctly” at the seventh Gathering for Gardner, in March 2006.

Click for Answer

December 15, 2016December 25, 2016 | Puzzles

Unquote

“To believe entails no desire to know; everybody reads the Bible, but who reads Flavius Josephus?” — Arthur Koestler

December 14, 2016 | Quotations

Black and White

hochberg puzzle

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?

	Select Click for Answer
The first rook goes on d2. Now if the king flees to the left, check him from the a-file, and if he flees to the right, check him from the g-file. So, for example, if he goes to e4, then play 2. g4+: That’s the key — this plan traps the king in a narrow pair of files, and now it’s an easy matter to attack those with the remaining rooks (above, for example: 2. … Kf3 3. Rf4+ 4. Ke3 Re2#). From Chess Braintwisters, 1999.

Click for Answer

December 14, 2016December 25, 2016 | Puzzles

Last Business

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.

December 13, 2016 | Death · Oddities

8 Is Enough

8 Is Enough - problem

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?

	Select Click for Answer
In long division, whenever two digits are brought down instead of one — as happens twice in this problem — a zero must appear in the quotient. That tells us that the quotient is x080x. The visible 8 multiplied by the divisor produces a three-digit number. But when the last number in the quotient is multiplied by the divisor, it produces a four-digit number. So the last digit in the quotient must be 9. The divisor has three digits and produces a three-digit number when multiplied by the visible 8. So the divisor must be less than 125, because 8 × 125 = 1000. That tells us something further. When the divisor is multiplied by the first digit of the quotient, it produces a three-digit number, and when this is subtracted from the first four digits of the dividend it yields a two-digit difference. That means that the first digit of the quotient must be greater than 7, because 7 × 124 = 868, and that would yield a three-digit difference when subtracted from even the smallest possible four-digit number, 1000 (1000 – 868 = 132). An 8 or a 9 could produce a two-digit difference, but a 9 would produce a four-digit number when multiplied by the divisor. So the first digit of the quotient is 8, and the full quotient is 80809. Now, 80809 × 123 = 9939507, which has only seven digits, and the dividend has eight. So the divisor is less than 125 and greater than 123, which means it’s 124. That gives us everything we need:

Click for Answer

December 13, 2016December 25, 2016 | Puzzles

Podcast Episode 133: Notes and Queries

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.

See full show notes …

December 12, 2016October 1, 2017 | Podcast

Kapitza’s Pendulum

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):

December 11, 2016December 11, 2016 | Science & Math

Small Business

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.)

December 10, 2016December 10, 2016 | Art

Memento Mori

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.

December 9, 2016December 8, 2016 | Art · Death