Drape a chain of evenly spaced weights over a pair of (frictionless) inclined planes like this. What will happen? There’s more mass on the left side, but the slope on the right side is steeper. Simon Stevin (1548-1620) realized that in fact the chain won’t move at all — if it did, we could link the ends as shown and produce a perpetual motion machine.
This is remembered as the “Epitaph of Stevinus.” Richard Feynman wrote, “If you get an epitaph like that on your gravestone, you are doing fine.”
The denominations of U.S. coins make intuitive sense, but they can be unwieldy: It can take up to eight coins to assemble an amount up through 99¢. Indeed, producing 99¢ takes (1 × 50¢) + (1 × 25¢) + (2 × 10¢) + (4 × 1¢). What five denominations would minimize the number of coins ever needed to make change?
In The Math Chat Book, Frank Morgan reports that with coins of 1¢, 3¢, 11¢, 27¢, and 34¢, you never need more than 5 coins to make change. For example, now 99¢ = (2 × 34¢) + (1 × 27¢) + (1 × 3¢) + (1 × 1¢). Of the 1,129 possible solutions, this one requires the fewest coins on average (3.343).
Unfortunately, this system is a bit tricky too — to assemble some totals, it’s more efficient to use a few middle-size coins rather than starting with the largest value possible. For example, if you assemble 54¢ by starting with a 34¢ coin, it takes four additional coins to gather the remaining 20¢: (1 × 11¢) + (3 × 3¢). It would have been simpler to choose 2 × 27¢, but that’s not immediately evident.
When we see Tom Hanks in a film, we think of him as a decent, honest everyman in part because we’ve seen him play decent, honest everymen in many other movies. Casting directors choose him in part for this reason — they know that the audience has established a sense of his persona from previous films, and that this affects their perception of him. We all know this; actors are hired deliberately to elicit these effects.
But a movie is fiction, and enjoying it requires restricting our attention to the fictional world in which it takes place. As experienced moviegoers, if we see the hero dangling from a cliff early in the film, we know that he’ll survive, but we repress this knowledge in order to enjoy the suspense that the filmmakers intend. We put our knowledge of movie lore on hold.
But isn’t this precisely the same sort of movie lore that we use when we let a star’s persona fill out the character he or she is playing?
“Why is it appropriate to put our knowledge of star personae to work when watching a movie, but not our knowledge of how popular plots go 99.9% of the time?” asks CUNY philosopher Noël Carroll. “Why is access to one kind of movie lore legitimate and access to the other kind not?”
(Noël Carroll, “The Problem With Movie Stars,” in his Minerva’s Night Out, 2013.)
Dance has a distinctive place among the performing arts. Dancers don’t “cause” a dance in the same way that musical instruments cause music. Rather, dancers are the dance — their movements instantiate it.
“You can’t describe a dance without talking about the dancer,” wrote American choreographer Merce Cunningham. “You can’t describe a dance that hasn’t been seen, and the way of seeing it has everything to do with the dancers.” A work of dance might be recorded abstractly in notation, but it’s the performance that realizes it; you can’t really encounter a dance without seeing it performed.
With that in mind, suppose that The Nutcracker is performed simultaneously in two different cities. If a dance work is fully realized only in performance, then can we really say that Performance A presents the same artwork as Performance B? If not, then what is The Nutcracker?
A related puzzle: Does a dance work last forever? It certainly has a beginning in time; does it have an end, if, say, it’s forgotten? Our species will one day become extinct — when that happens, will The Nutcracker cease to exist?
(Jenny Bunker, et al., Thinking Through Dance, 2013; Graham McFee, The Philosophical Aesthetics of Dance, 2011.)
emptitious
adj. susceptible to bribery
chantage
n. extortion; blackmail
luition
n. payment of a ransom
quomodocunquizing
adj. that makes money in any possible way
Ivan Konstantinovich Aivazovsky (1817-1900) earned fame throughout Russia for his astonishingly realistic seascapes, which capture the expressive quality of ocean waters, and in particular the play of sunlight and moonlight on surging waves. More than half of the artist’s 6,000 canvases are devoted to his fascination with moving water.
Remarkably, these were painted from memory, far from the sea. “We can perfectly well understand that when he painted The Ninth Wave or The Wreck, he had no need to watch the ever-shifting colour and movement of the great waters as he worked, for these pictures are poems in which the artist has concentrated an amplitude of observation and experience,” wrote Rosa Newmarch in 1917. “We realize that their impressive, haunting grandeur is no more spontaneous than the impressiveness of many a great sonnet; they are rather the aftermath of his passion for the sea.”
His successes made him equally popular among the people and among his fellow artists. Ivan Kramskoi wrote, “Aivazovsky is — no matter who says what — a star of first magnitude, and not only in our [country], but also in history of art in general.” And the saying “worthy of Aivazovsky’s brush” was used in Russia to describe anything ineffably lovely.
Wikimedia Commons has a collection of his seascapes.
During a stickup, bank robbers order tellers to keep their hands up so they can’t defend themselves or the customers. In 1921 San Francisco inventor Harry McGrath offered this solution: The teller wears a loaded pistol under his arm, with a wire running down his coat sleeve to his palm. Now when his arms are raised he can still fire the gun.
The patent says nothing about aiming, but “in order to make the gun perfectly safe, a blank cartridge can be placed in the magazine to be fired first, followed by a ball cartridge.”
I don’t know whether McGrath himself was a bank teller. I hope not.
A puzzle by Henry Dudeney. Frogs sit on eight of these 64 tumblers so that no two occupy the same row, column, or diagonal. “The puzzle is this. Three of the frogs are supposed to jump from their present position to three vacant glasses, so that in their new relative positions still no two frogs shall be in a line. What are the jumps made?” The frogs may not exchange positions; each must jump to a glass that was not previously occupied.
(“But surely there must be scores of solutions?” “I shall be very glad if you can find them. I only know of one — or rather two, counting a reversal, which occurs in consequence of the position being symmetrical.”)
“Little minds are interested in the extraordinary; great minds in the commonplace.” — Elbert Hubbard
In 1616, British officer Nathaniel Courthope was sent to a tiny island in the East Indies to contest a Dutch monopoly on nutmeg. He and his men would spend four years battling sickness, starvation, and enemy attacks to defend the island’s bounty. In this week’s episode of the Futility Closet podcast we’ll describe Courthope’s stand and its surprising impact in world history.
We’ll also meet a Serbian hermit and puzzle over an unusual business strategy.
