Express

This paragraph jumped out at me last night from B.G. Wilson’s Unusual Railways (1958) — he’s writing about the Mount Washington Cog Railway in New Hampshire, one of the world’s steepest, with an average grade of 25 percent:

Before leaving this line, mention must be made of a method of riding down the track employed by track maintenance men and long since banned. Wood and metal seats some 3 ft. × 1 ft. were made to fit over the rack rail. These were known as slide-boards, or more popularly, as ‘Devil’s Shingles’. Seated on these, controlling (sometimes) the speed with hand brakes, the men would career down the mountainside. The record time for the trip — as we have said, 3 1/4 miles — was 2 3/4 minutes!

That’s 70 mph! For comparison, the modern train takes 40 minutes to descend at 4.6 mph. And this was in the late 19th century — the railway opened in August 1869. Wilson writes, “After one man had been killed and another seriously injured, the Devil’s Shingles were banned.” I don’t know any more than that.

December 19, 2015December 18, 2015 | Oddities · Technology

All Hands on Deck?

A reader named Hamp Stevens sent this conundrum to Martin Gardner, who published it in his Mathematical Magic Show (1965). Can these 25 cards be arranged to form five poker hands, each of them a straight or better (that is, straight, flush, full house, four of a kind, straight flush, or royal flush)? If it’s possible, find the five hands; if it’s not, prove that it’s impossible.

“This ingenious puzzle is quickly solved if you go about it correctly,” Gardner wrote. “A single card is the key.”

	Select Click for Answer
The key is the 4 of hearts. There is no 3 or 5 present, so the 4 of hearts can’t be part of a straight. There are only three other hearts, so it can’t be part of a flush. It’s the only 4 in the set, so it can’t be part of a full house. And the set contains no four cards of the same value, so the four of hearts can’t be the fifth card accompanying four of a kind.

December 18, 2015January 3, 2016 | Puzzles

Fitting

arc lengths

If two unit circles are tangent externally as shown, and from a point P on one circle rays PQ and PR are drawn intersecting both circles, then arc lengths x + y = z.

From Claudi Alsina and Roger B. Nelsen, Icons of Mathematics, 2011.

December 17, 2015December 15, 2015 | Science & Math

Black and White

carpenter chess problem

By George Edward Carpenter. White to mate in two moves.

	Select Click for Answer
1. Qh3! Ke4 2. Rc4# From the Dubuque Chess Journal, 1873.

December 17, 2015December 27, 2015 | Puzzles

Hard Target

In a special football game, a team scores 7 points for a touchdown and 3 points for a field goal. What’s the largest mathematically unreachable number of points that a team can score (in an infinitely long game)?

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Eleven. For any integer n greater than that, the possible remainders when n is divided by 3 are 0, 1, and 2. If the remainder is 0, then we can get to n by scoring field goals. If the remainder is 1, then n – 13 has remainder 0 and we can reach n by scoring one touchdown and some number of field goals. And if the remainder is 2, then n – 14 has remainder 0 and we can get to n by scoring two touchdowns and some number of field goals. Put simply, all integers n greater than 11 can be written in the form n = 7a + 3b for some nonnegative integers a and b.

Click for Answer

December 16, 2015December 27, 2015 | Puzzles

Unquote

la rochefoucauld

More maxims of François VI, Duc de La Rochefoucauld (1613–1680):

“We always love those who admire us; but we don’t always love those whom we admire.”
“There are people who would never have been in love, if they had never heard talk of Love.”
“The Generality of People judge of Men by their Reputation, or Fortune.”
“Men would not live long in Society, if they were not the mutual Dupes of one another.”
“Titles, instead of exalting, debase those who don’t act up to them.”
“Prosperity is a stronger Trial of Virtue than Adversity.”
“Weak People can’t be sincere.”
“‘Tis more difficult to be faithful to a Mistress when on good Terms with her, than when on bad.”
“‘Tis not so dangerous to do Ill to most Men as to do them too much Good.”
“A Man often imagines he acts, when he is acted upon; and while his Mind aims at one thing, his Heart insensibly gravitates towards another.”
“When great Men suffer themselves to be subdued by the Length of their Misfortunes, they discover that the Strength of their Ambition, not of their Understanding, was what supported them; and that, bating a little Vanity, Heroes are just like other Men.”
“Cunning and Treachery proceed from Want of Capacity.”
“If we took as much Pains to be what we ought, as we do to deceive others by disguising what we are; we might appear as we are, without being at the Trouble of any Disguise.”

And “‘Tis a Mistake to imagine that only the violent Passions, such as Ambition and Love, can triumph over the rest. Laziness, languid as it is, often masters them all; she indeed influences all our Designs and Actions, and insensibly consumes and destroys both the Passions and the Virtues.”

December 16, 2015December 15, 2015 | Quotations

Pleased, I’m Sure

A problem by Atlantic College mathematician Paul Belcher:

Anna and Bert invite n other couples to a dinner party. Before the meal begins, some people shake hands. No one shakes hands with their own partner, no one shakes hands with themselves, and no two people shake hands with each other more than once. Afterward, Anna asks all the other 2n + 1 people how many times they shook hands, and she gets a different answer from each of them. How many times did Anna shake hands?

	Select Click for Answer
Altogether there are 2n + 2 people at the party. None shakes hands with themselves or their partner, so the maximum number of handshakes that one person could have participated in is 2n, and the minimum is 0. Anna receives a different answer from each of the 2n + 1 people she asks, so the answers she gets must be 0, 1, 2, …, 2n. Let the person who shook hands 2n times be called Max and the person who shook hands 0 times be called Minie (making no assumptions about gender). Max must have shaken hands with everyone but his or her own partner. Since Max has shaken hands with Anna, Max cannot be Bert and Max’s partner must be Minie. Now remove Max and Minie and all their handshakes. This decreases by 1 the number of handshakes that each of the others participated in, and leaves us facing essentially the same problem but with n – 1 other couples. If we repeat this process, eventually we’re left with Anna and Bert as the only two people remaining. In that situation, neither of them shakes hands. And since one handshake disappeared as each of the other n couples was removed from the problem, both Anna and Bert have shaken hands n times. (Paul Belcher, “But What Colour Are Anna’s Eyes?”, Mathematical Gazette 87:509 [July 2003], 322-323.)

Click for Answer

December 15, 2015December 27, 2015 | Puzzles

In a Word

topolatry
n. excessive reverence for a place

Of the million or so Japanese who visit Paris each year, about 12 have to be repatriated due to “Paris syndrome,” a transient psychological disorder brought on when the mundane reality of the city clashes with their romanticized expectations.

The syndrome was first diagnosed by Hiroaki Ota, a Japanese psychiatrist working in France. Symptoms include delusional states, hallucinations, feelings of persecution, and anxiety.

“Fragile travellers can lose their bearings,” psychologist Hervé Benhamou told Le Journal du Dimanche. “When the idea they have of the country meets the reality of what they discover, it can provoke a crisis.”

(A. Viala et al., “Les japonais en voyage pathologique à Paris : un modèle original de prise en charge transculturelle,” Nervure 5 (2004): 31–34.)

December 14, 2015 | Oddities

Podcast Episode 85: Raising Chicago

In 1868, visiting Scotsman David Macrae was astonished to see Chicago transforming itself — dozens of buildings were transplanted to the suburbs, and hotels weighing hundreds of tons were raised on jackscrews. In this week’s episode of the Futility Closet podcast we’ll follow the city’s astounding 20-year effort to rid itself of sewage and disease.

We’ll also learn how a bear almost started World War III and puzzle over the importance of a ringing phone.

See full show notes …

December 14, 2015October 19, 2017 | History · Podcast · Technology

Putting Words

In 1946, when Japanese prime minister Hideki Tojo was being held prisoner by the victorious Allies, he asked for a set of dentures so that he could speak clearly during his war crimes trial.

The dentures were made by 22-year-old military dentist E.J. Mallory. “I figured it was my duty to carry out the assignment,” Mallory remembered in 1988. “But that didn’t mean I couldn’t have fun with it.”

An amateur ham radio operator, he inscribed the phrase “Remember Pearl Harbor” in Morse code into the dentures and delivered them to Tojo.

Mallory and his colleague George Foster told a few friends, but the secret got out and the two had to awaken Tojo in the middle of the night to borrow back the dentures and grind out the message. The next day, when a colonel confronted them, they were able to say truthfully that there was no message.

It’s not known whether Tojo ever found out what had happened. He was executed in 1948.

“It wasn’t anything done in anger,” Mallory remembered in 1995. “It’s just that not many people had the chance to get those words into his mouth.”

December 13, 2015December 12, 2015 | History · Oddities