A puzzle from the excellent Riddler feature at FiveThirtyEight, via Oliver Roeder’s 2018 collection The Riddler:
Your eccentric friend Nigel flies from Heathrow to an airport somewhere in the 48 contiguous states, then hires a car and drives around the country, touching the Atlantic and Pacific Oceans and the Gulf of Mexico, then returns to the airport at which he started and flies home. If he crossed the Ohio River once, the Missouri River twice, the Mississippi River three times, and the Continental Divide four times, then there’s one state that we can say for certain that he visited on his trip. What is it?
Minnesota. In order to cross a river an odd number of times and still return to the original airport, Nigel must “go around” the river somehow. That’s not hard with the Ohio, but the Mississippi is enormous and practically divides the country. So in order to cross it three times and return to his airport, at some point he’d have had to go around the source at Lake Itasca, Minnesota.
Korf’s clock is of a novel sort
In which two pairs of hands are used:
One pair points forwards as it ought,
The other backwards a la Proust.
When it says eight it’s also four,
When it says nine it’s also three;
A single glance and you no more
Need fear the ancient Enemy.
For with this wondrous clock you’ll find
As, Janus-like, it turns about
(To such an end it was designed)
Time simply cancels itself out.
Palmström’s Clock
But Palmström’s clock has a “higher” power,
Balanced as lightly as a flower.
Scorning a set pedestrian pace,
It keeps time with a certain grace
And will, in answer to a prayer,
Go en retard, en arriéré.
One hour, two hours, three hours indeed,
Sympathizing with our need!
Though clockwork in its outward part
It hides within — a tender heart.
— Christian Morgenstern
Above: Built in 1586, the town hall in the old Jewish ghetto of Prague bears two clocks: a traditional clock tower with four faces bearing Roman numerals and a second clock bearing Hebrew numerals. The hands on the conventional clocks turn clockwise; those on the Hebrew clock turn counterclockwise. (Thanks, Danesh.)
A puzzle from James F. Fixx’s More Games for the Superintelligent, 1976:
A man who likes trains walks occasionally to a nearby railroad track and waits for one to go by. Afterward he notes whether he saw a passenger train or a freight. After several years his notes show that 90 percent of the trains he’s seen have been passenger trains. One day he meets an official of the railroad and is surprised to learn that the passenger and freight trains on this line are precisely equal in number. If the man timed his trips to the track at random, why did he see such a disproportionate number of passenger trains?
All the trains run hourly, but the freight trains follow six minutes after the passenger trains. So if the man arrives at the track at random, his odds of arriving after a freight train and before a passenger train are 9 to 1; for 54 out of every 60 minutes, a passenger train will be the next to pass.
In 1964, sociolinguist William Labov ran a revealing experiment in three New York department stores, Saks Fifth Avenue, Macy’s, and S. Klein. Of the three, Saks generally commanded the highest prestige and S. Klein the lowest. Labov had found that one marker of social stratification in the city was the pronunciation of the letter R, and he wanted to see whether this was reflected in the speech of the salespeople at the various stores.
He did this by approaching a salesperson in each store and asking directions to a department on the fourth floor. When the salesperson told him “Fourth floor,” he leaned forward and said, “Excuse me?” This forced the person to say the phrase “Fourth floor” again, this time rather self-consciously.
As expected, Labov found that salespeople at the upscale Saks tended to pronounce their Rs, while those at the lower-priced Klein tended to the broader New York pronunciation “fawth flaw.” But when asked to repeat the phrase, those at Macy’s and Klein’s tended to amend their pronunciation to sound more “classy.”
“How can we account for the differences between Saks and Macy’s?” Labov wrote. “I think we can say this: the shift from the influence of the New England prestige pattern [r-less] to the mid-Western prestige pattern [r-full] is felt most completely at Saks. The young people at Saks are under the influence of the r-pronouncing pattern, and the older ones are not. At Macy’s there is less sensitivity to the effect among a large number of younger speakers who are completely immersed in the New York City linguistic tradition. The stockboys, the young salesgirls, are not as yet fully aware of the prestige attached to r-pronunciation. On the other hand, the older people at Macy’s tend to adopt this pronunciation: very few of them rely upon the older pattern of prestige pronunciation which supports the r-less tendency of older Saks sales people.”
In separate interviews Labov found that two thirds of New Yorkers felt that outsiders disliked the city accent. “They think we’re all murderers,” one man told him. A woman said, “To be recognized as a New Yorker — that would be a terrible slap in the face.”
(William Labov, The Social Stratification of English in New York City, 2006.)
A brainteaser from the Soviet science magazine Kvant, via Quantum, January/February 1991:
Bobby found the sum of three consecutive integers, then of the next three consecutive integers, then multiplied these two sums together. Could the product have been 111,111,111?
No. One of the two groups must contain one even number and two odd ones. The sum of these three numbers must be even, so the product of the two sums must be even as well.
William Wallace Cook (1867-1933) claimed to have worn out 25 typewriters in as many years turning out hundreds of nickel and dime novels, all of them written in the same format, 40,000 words divided into 16 chapters of five single-spaced pages each. At the end of his career he published his system for generating plots, billed as “Plotto, an invention which reduces literature to an exact science.”
The “invention” is really a list of story ideas, all molded to Cook’s basic notion of a plot: “Purpose, opposed by Obstacle, yields Conflict.” The protagonist wants to find happiness in love and courtship, married life, or enterprise; he encounters a conflict and must reach a resolution. What makes the book fun is the absurd specificity of some of the ideas. Here’s an example:
1367
(b) (1083)(1287)
A has invented a life preserver for the use of shipwrecked persons*
A, in order to prove the value of the life preserver he has invented, dons the rubber suit, inflates it and secretly, by night, drops overboard from a steamer on the high seas.** (1414b) (1419b)
The numbers refer to elements that might be varied, to related plots, and to character types that might figure in the story. Varying the combinations might produce several million different stories. This is certainly formulaic, but, Cook said, “There are any number of highbrow authors who will ridicule this invention in public and use it in private.” (In fact both Alfred Hitchcock and Erle Stanley Gardner admitted in interviews that they’d read the book, which went through multiple editions.)
The numbered master list gives 1,462 plots, all linked with character symbols and apparently all thought up by the author. The full text is on the Internet Archive.
Albrecht Dürer’s 1514 engraving Melencolia I includes this famous magic square: The magic sum of 34 can be reached by adding the numbers in any row, column, diagonal, or quadrant; the four center squares; the four corner squares; the four numbers clockwise from the corners; or the four counterclockwise.
In Power Play (1997), University of Toronto mathematician Ed Barbeau points out that there’s even more magic when we consider squares and cubes. Take the numbers in the first two and the last two rows:
In his 2014 book Describing Gods, Graham Oppy presents the “divine liar” paradox, by SUNY philosopher Patrick Grim:
1. X believes that (1) is not true.
If we suppose that (1) is true, then this tells us that X believes that (1) is not true. But if an omniscient being believes that (1) is not true, then it follows that (1) is not true. So the assumption that (1) is true leads to a contradiction.
Suppose instead that (1) is not true. That is, suppose that it’s not the case that X believes that (1) is not true. If an omniscient being fails to believe that (1) is not true, then it’s not true that (1) is not true. So this alternative also leads to a contradiction.
But, on the assumption that there is an omniscient being X, either it’s the case that (1) is true or it’s the case that (1) is not true.
“So, on pain of contradiction,” Oppy explains, “we seem driven to the conclusion that there is no omniscient being X.”
(Also: Patrick Grim, “Some Neglected Problems of Omniscience,” American Philosophical Quarterly 20:3 [July 1983], 265-276.)
The lost art of floriography assigned meanings to flowers so that lovers could exchange messages with “talking bouquets.” In his 1839 Language of Flowers, English journalist Frederic Shoberl rendered an entire verse by French poet Évariste de Parny as the combination of 16 flowers:
Aimer est un destin charmant,
C’est un bonheur qui nous enivre,
Et qui produit l’enchantement.
Avoir aimé, c’est ne plus vivre,
Hélas! c’est avoir acheté
Cette accablante vérité,
Que les serments sont un mensonge,
Que l’amour trompe tôt ou tard,
Que l’innocence n’est qu’un art,
Et que le bonheur n’est qu’un songe.
“It may be thus rendered: ‘To love is a pleasure, a happiness, which intoxicates; to love no longer, is to live no longer; it is to have bought this sad truth, that innocence is falsehood, that love is an art, and that happiness is a dream.'”
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