Table and Tumblers

This problem originated in Russia, according to various sources, but no one’s sure precisely where:

Before you is a square table that can rotate freely. In each corner is a deep well, at the bottom of which is a tumbler that’s either upright or inverted. You can’t see the tumblers, but you can reach into the wells to feel their positions.

Periodically the table rotates and stops at random. After each stop, you can feel two of the tumblers and turn over either, both, or neither. If all four of the tumblers are in the same state — all upright or all inverted — then a bell sounds. Otherwise the table rotates again and you make another “move.”

Can you guarantee to ring the bell in a finite number of moves? If so, how?

Click for Answer
| Puzzles

The Wheels of Justice

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In England’s Court of Chancery, a litigant’s charges were converted into “interrogatories,” or searching questions to be put to the defendant, who had to answer them under oath. For example:

“Whether or no was the said testator, G. H., at the time of his death indebted to any and what persons or person in any and what sums or sum of money?”

Composing these questions was such mournfully mechanical work that a thoughtless clerk could turn almost anything into an interrogatory. Junior counsel Edward Karslake once submitted a rather florid account of a broken trust, and the clerk absently returned this:

“Did not the defendant fall down on her knees or on one and which of them and implore the plaintiff with tears in her eyes or in one and which of them to advance the said sum of £—- to her husband to save him from bankruptcy and their children from ruin or how otherwise?”

According to one story, a junior counsel wagered that if the first few lines of Paradise Lost were inserted into a bill, his clerk would render them into interrogatories “without turning a hair.” Here’s Milton’s text:

Of Man’s First Disobedience, and the Fruit
Of that Forbidden Tree, whose mortal taste
Brought Death into the World, and all our woe …

And here’s what the clerk produced:

“Was it man’s first or some other and what disobedience, and the fruit of that forbidden or some other and what tree, whose mortal taste brought death into this or some other and what world and all our woe, and if not why not or how otherwise?”

It was this tedious officiousness that Dickens railed about in Bleak House. He had some justice: The novel’s central case was inspired by a Chancery suit that took 36 years to get through court.

| Society

Wanted

If you see a kangaroo wearing a man’s waistcoat with $20 in one pocket, please notify William Thompson, farmer of Grafton, near Sydney, Australia. When Thompson found a kangaroo caught in a wire fence he acted on impulse, removing his old waistcoat and buttoning it firmly on the kangaroo, which then bounded away. It was several hours later when he remembered the money in the pocket.

— The Paris News, Feb. 4, 1946

| Oddities

Exercises

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One summer afternoon in 1917, Royal Flying Corps trainee Graham Donald prepared to try a new maneuver with his Sopwith Camel. He ascended into a vertical loop, intending to flip the plane at the top and fly off in the opposite direction. Unfortunately, when the airplane was fully inverted at 6,000 feet, his safety belt gave way and “suddenly I dived clean through it and fell out of the cockpit.”

“The first 2,000 feet passed very quickly, and terra firma looked damnably ‘firma,'” he recalled later. But as he fell, “I began to hear my faithful little Camel somewhere nearby.” He dropped onto the diving plane and managed to grip its top wing, “and that saved me from slithering straight through the propeller, which was glistening beautifully in the evening sunshine.”

As the ground neared at 140 mph, he reached into the cockpit and pulled back on the control stick. Unfortunately, this sent the plane into an inverted spin. With 2,500 feet left, Donald managed to put his right foot on the stick and push it forward, and he found himself clinging to a plane that was flying upside down. He reached the controls, righted the plane, and climbed into the cockpit with about 800 feet to spare. To prevent further strain on the wings, he cut the engine and glided back to the airfield.

“I made an unusually good landing, but there was no one there to applaud — every man-jack of the squadron had mysteriously disappeared. After a minute or so, heads began to appear all over the place — popping up like bunny rabbits from every hole. Apparently, when I had pressed my foot on the control stick, I’d also pressed both triggers and the entire airfield had been sprinkled with bullets. Very wisely, the ground crew dived as one man for the nearest ditch.”

(From Joshua Levine’s 2008 book On a Wing and a Prayer. Thanks, Paul.)

| Oddities

Pastorale

A logic puzzle by Lewis Carroll, July 2, 1893. What conclusion can be drawn from these premises?

  1. All who neither dance on tight-ropes nor eat penny-buns are old.
  2. Pigs that are liable to giddiness are treated with respect.
  3. A wise balloonist takes an umbrella with him.
  4. No one ought to lunch in public who looks ridiculous and eats penny-buns.
  5. Young creatures who go up in balloons are liable to giddiness.
  6. Fat creatures who look ridiculous may lunch in public, provided that they do not dance on tight-ropes.
  7. No wise creatures dance on tight-ropes if liable to giddiness.
  8. A pig looks ridiculous carrying an umbrella.
  9. All who do not dance on tight-ropes and who are treated with respect are fat.
Click for Answer
| Puzzles

“A Compulsion to Teach”

Marvin Harold Hewitt was bright enough to find schoolwork boring, so he dropped out of high school. That didn’t hold him back:

  • To get a job at 23, he claimed he was a Temple University undergraduate and found himself teaching arithmetic, geography, and history at Camp Hill Military Academy in Philadelphia.
  • Discovering a taste for teaching, he borrowed the identity of Columbia physicist Julius Ashkin and began teaching calculus, algebra, and trigonometry at the Philadelphia College of Pharmacy and Science, where he also bluffed his way through managing a physics lab.
  • Still posing as Ashkin, he taught analytical and solid geometry, algebra, and physics at the Bemidji State Teachers College in Minnesota; moved to St. Louis University, where he taught graduate courses in nuclear physics, statistical mechanics, thermodynamics, and tensor analysis; and arrived finally at the University of Utah, which hired him as a full professor of physics.
  • Here his luck finally ran out — a number of people noticed discrepancies in his stories, and he had to admit his real identity and return to his mother in Philadelphia.
  • Chastened but not discouraged, he applied next to teach at the University of Arkansas College of Engineering, posing as “George Hewitt,” a former research director of RCA. They hired him, but a former RCA employee exposed him the following spring and he was fired.
  • Posing as “Clifford Berry,” he took a job teaching second- and third-year physics and calculus at the New York State Maritime College in the Bronx.
  • Posing as “Kenneth P. Yates,” he taught physics at the University of New Hampshire at Durham, where he was finally outed by a suspicious student amid publicity too wide to live down.

In all, Hewitt spent nine years teaching mathematics, engineering, and physics at seven different schools and universities, using forged credentials throughout. “The ease with which Hewitt obtained these jobs fills him with indignation,” reported Life magazine, which profiled him in 1954 when it was all over. “The unquestioning acceptance of a transcript and careless checking of references is, in his fairly expert opinion, a universal weakness throughout the U.S. higher educational system. When he considers what might have happened to a great many people had he made medicine or surgery his field, he shudders.”

| Hoaxes

The Less Said

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What’s the most comprehensive contraction? Philip Cohen proposes this story:

An old salt was telling of going through a typhoon in his sailing ship. At the top of the storm, he said, ‘M’ jibs’l’s lines snapped. And m’t’g’ll’nts’ls’d’a done the same if it hadn’t slacked off just then.’

M’t’g’ll’nts’ls’d’a means “my topgallant sails would have” — a savings of 14 letters and four spaces using seven apostrophes. An old sailor knows the value of efficiency.

| Language

Rolling

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Suppose we set a small circle rolling around the interior of a large circle of twice its diameter. If we follow a point on the small circle, what pattern will it draw?

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| Science & Math

“A Unique Will”

At a dinner for law alumni of New York University in 1907, Walter Lloyd Smith of the New York Supreme Court read “the most remarkable document that ever came into his possession” — the will of an inmate of the Cook County Insane Asylum at Dunning, Ill.:

I, Charles Lounsbury, being of sound mind and disposing memory, do hereby make and publish this, my last will and testament, in order as justly as may be to distribute my interest in the world among succeeding men.

That part of my interest which is known in law and recognized in the sheep-bound volumes as my property, being inconsiderable and of no account, I make no disposal of in this my will.

My right to live, being but a life estate, is not at my disposal, but these things excepted all else in the world I now proceed to devise and bequeath.

Item: I give to good fathers and mothers, in trust for their children, all good little words of praise and encouragement, and all quaint pet names and endearments, and I charge said parents to use them justly and generously, as the needs of their children may require.

Item: I leave to children inclusively, but only for the term of their childhood, all and every, the flowers of the fields, and the blossoms of the woods, with the right to play among them freely according to the customs of children, warning them at the same time against thistles and thorns. And I devise to children the banks of the brooks, and the golden sands beneath the waters thereof, and the odors of the willows that dip therein, and the white clouds that float high over the giant trees. And I leave the children the long, long days to be merry in, in a thousand ways, and the night and the moon and the train of the Milky Way to wonder at, but subject nevertheless to the rights hereinafter given to lovers.

Item: I devise to boys jointly all the useful idle fields and commons where ball may be played; all pleasant waters where one may swim; all snowclad hills where one may coast, and all streams and ponds where one may fish, or where, when grim Winter comes, one may skate; to have and to hold the same for the period of their boyhood. And all meadows with the clover blossoms and butterflies thereof, the woods and their appurtenances, the squirrels and the birds, and echoes and strange noises, and all distant places which may be visited, together with the adventures there found. And I give to said boys each his own place at the fireside at night, with all pictures that may be seen in the burning wood, to enjoy without let or hindrance and without any incumbrance of care.

Item: To lovers I devise their imaginary world with whatever they may need; as the stars of the sky; the red roses by the wall; the bloom of the hawthorn; the sweet strains of music, and aught else by which they may desire to figure to each others the lastingness and beauty of their love.

Item: To young men jointly, I devise and bequeath all boisterous, inspiring sports of rivalry, and I give to them the disdain of weakness and undaunted confidence in their own strength, though they are rude; I give them the power to make lasting friendships, and of possessing companions, and to them exclusively I give all merry songs and brave choruses, to sing with lusty voices.

Item: And to those who are no longer children or youths or lovers, I leave memory, and I bequeath to them the volumes of the poems of Burns and Shakespeare and of other poets, if there be others, to the end that they may live over the old days again, freely and fully, without tithe or diminution.

Item: To our loved ones with snowy crowns I bequeath the happiness of old age, the love and gratitude of their children until they fall asleep.

The original, it turns out, was written by Williston Fish in 1897 and published in Harper’s Weekly the following year. He had intended it as a poetic trifle, but newspapers around the country had picked it up and run it as fact, often embellishing the language, until, Fish wrote in 1908, “this one of my pieces has been translated into all the idiot tongues of English.” Charles Lounsbury was the name of an old relative of his — “a big, strong all-around good kind of man,” but not, evidently, insane.

| Oddities