The Unheard Islander

A puzzle by Edward J. Barbeau, from the February 2007 issue of Crux Mathematicorum:

A certain familiar island is inhabited by knights, who can only speak the truth, and knaves, who can only lie. One day a visitor meets three inhabitants, A, B, and C. The visitor asked, “How many knights are there among you three?”

A gave an answer, which the visitor didn’t hear. When the visitor asked B what A had said, B replied, “A said that there is one knight among us.” At this C said, “Don’t believe B. He is lying.”

What are B and C?

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The Region Beta Paradox
Image: Wikimedia Commons

Suppose you decide that you’ll walk (at 3 mph) to any destination that’s within a mile of your house, and bike (at 15 mph) to any destination that’s farther away. That’s a reasonable choice, but it has a surprising result: You’ll actually arrive more quickly at moderately distant points (1 to 5 miles away) than at most points closer to home (less than 1 mile away).

Psychologist Daniel Gilbert uses this example to illustrate a phenomenon in our reactions to stressful events: Sometimes we’ll recover more quickly from particularly distressing experiences because they’re strong enough to invoke defense processes that attentuate stress.

The Six Submarines

A puzzle by Henry Dudeney:

If five submarines, sunk on the same day, all went down at the same spot where another had previously been sunk, how might they all lie at rest so that every one of the six U-boats should touch every other one? To simplify we will say, place six ordinary wooden matches so that every match shall touch every other match. No bending or breaking allowed.

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J.H. Brown’s 1864 book Spectropia: Or, Surprising Spectral Illusions promises to show “ghosts everywhere, and of any colour.” It accomplishes this by relying on two simple principles: persistence of vision and complementary colors. Readers are directed to stare at any of the figures for 15 seconds and then turn their eyes to a white surface (or the sky); “the spectre will soon begin to make its appearance, increasing in intensity, and then gradually vanishing,” in the color complementary to that of the stimulus.

Try it yourself.

All Together Now

Kurt Vonnegut wrote, “The function of the artist is to make people like life better than before.”

Asked whether he’d ever seen this done, he said, “Yes, the Beatles did it.”

(From Dan Wakefield’s introduction to Vonnegut’s If This Isn’t Nice, What Is?, 2013.)


After Archimedes’ death in 212 B.C., his tomb in Sicily fell into obscurity and was eventually lost. It was rediscovered by, of all people, Cicero, who had been sent to the island in 75 B.C. to administer corn production:

When I was Quaestor, I tracked down his grave; the Syracusans not only had no idea where it was, they denied it even existed. I found it surrounded and covered by brambles and thickets. I remembered that some lines of doggerel I had heard were inscribed on his tomb to the effect that a sphere and a cylinder had been placed on its top. So I took a good look around (for there are a lot of graves at the Agrigentine Gate cemetery) and noticed a small column rising a little way above some bushes, on which stood a sphere and a cylinder. I immediately told the Syracusans (some of their leading men were with me) that I thought I had found what I was looking for. Slaves were sent in with scythes to clear the ground and once a path had been opened up we approached the pedestal. About half the lines of the epigram were still legible although the rest had worn away.

“So, you see, one of the most celebrated cities of Greece, once upon a time a great seat of learning too, would have been ignorant of the grave of one of its most intellectually gifted citizens — had it not been for a man from Arpinum who pointed it out to them.”

(From Anthony Everitt, Cicero, 2003.)

Image: Royal Academy of Music

This is charming: Looking for a place to practice her drawing one day, an anonymous child in the 1700s chose the blank spaces in a music book. In doing so, she made herself immortal, as the music is now held in university collections.

The figure at left is scrawled in John Wilbye’s Second Set of Madrigales, now at the Royal Academy of Music; the one on the right is in Thomas Weelkes’ Balletts and Madrigals at Harvard. Drawings and handwriting exercises apparently by the same child appear in music books at UCLA, Huntington Library, and the University of Illinois. (The drawings all appear in tenor books, which suggests that these copies had been bound in one volume when the girl drew the pictures.)

The child’s identity is unknown, although the name Eliza Richardson accompanies one practice alphabet. She seems to be drawing the same woman consistently, recognizable by her prominent nose, strong chin, and thick neck. It’s not clear when the drawings were made, but the woman appears to be wearing a sack-back gown, a style that was most popular between 1720 and 1770. According to Durham University music professor David Greer, “The dense patterns, small cap, and closely dressed hairstyle suggest the early part of this period.” But I believe that’s all we know.

(David Greer, Manuscript Inscriptions in Early English Printed Music, 2015.)

“Prisons for Animals”

This short composition was discovered in the Stanford University Library, tipped into the inside cover of a bound first volume of Charlotte Perkins Gilman’s 1909 magazine The Forerunner:

Spring is in the air. All creatures feel it. The fish are shooting up the rivers, the birds hard working and happy; every animal feels the lift and stir and new life. Even those which are in prison. …

What excuse has the Prison for Animals? What have they done to merit this life sentence?

Spring is in the air. The trees are misty with soft color, blurred with swelling buds, all aslant with curly tassels of young blossoms. The grass is pushing up in joyous vigor, green as it is never green again; soft, sweet, the delicious new first growth; beginning of a long summer’s feasting.

Here are the deer prisons. They have a high iron fence around them, another railing outside that. They have a wooden house for shelter. They have underfoot, cinders — gravel and cinders. …

To keep in a prison yard an animal built for speed, accustomed to wide ranging, to long swift flight, is cruelty. …

And for what? For whose benefit? Does it give pleasure? Those who find pleasure in gazing at helpless pain had better go unpleased. …

These beasts in prison, these who bear no burdens, provide neither food nor drink, wool nor hide — what excuse have we for tormenting them?

Here is a bald eagle. A bird of freedom. … The eagle sits huddled, dull as a brooding vulture. …

Here is a hawk, fierce-eyed. He beats his wings to tatters … against the bars.

Here is an elephant, huge, patient, with small, smouldering eyes that see more than we think. Manacled, this beast, chained at both ends, fore foot and hind foot, to stout posts. The elephant is a water lover. His dry hide itches for water. He wants to wade into it, to draw it up and pour it all over himself. …

All wild creatures have a keen, delicate sense of smell … We imprison them in fetid odors. They needs must breathe, night and day, the repulsive smell of their enemies, odors of danger and distrust. …

(Via Robert Alexander, ed., Spring Phantoms, 2018.)