Cyclic Billiards

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A puzzle from Colin White’s Projectile Dynamics in Sport (2010): Suppose a billiard table has a length twice its width and that a rolling ball loses no energy to bounces or friction but simply caroms around the table forever. Call the angle between the launch direction and the long side of the table α. At what angle(s) should the ball be hit so that it will arrive back at the same point on the table and traveling in the same direction, so that its motion is cyclic, following the same path repeatedly?

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Podcast Episode 115: Gettysburg’s Unknown Soldier

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After the Battle of Gettysburg, a dead Union soldier was found near the center of town. He bore no identification, but in his hands he held a photograph of three children. In this week’s episode of the Futility Closet podcast we’ll follow the efforts of one Philadelphia physician to track down the lost man’s family using only the image of his children.

We’ll also sample a 9-year-old’s comedy of manners and puzzle over a letter that copies itself.

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Podcast Episode 113: The Battle Over Mother’s Day

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Anna Jarvis organized the first observance of Mother’s Day in 1908 and campaigned to have the holiday adopted throughout the country. But her next four decades were filled with bitterness and acrimony as she watched her “holy day” devolve into a “burdensome, wasteful, expensive gift-day.” In this week’s episode of the Futility Closet podcast, we’ll follow the evolution of Mother’s Day and Jarvis’ belligerent efforts to control it.

We’ll also meet a dog that flummoxed the Nazis and puzzle over why a man is fired for doing his job too well.

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Podcast Episode 111: Japanese Fire Balloons

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Toward the end of World War II, Japan launched a strange new attack on the United States: thousands of paper balloons that would sail 5,000 miles to drop bombs on the American mainland. In this week’s episode of the Futility Closet podcast, we’ll tell the curious story of the Japanese fire balloons, the world’s first intercontinental weapon.

We’ll also discuss how to tell time by cannon and puzzle over how to find a lost tortoise.

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Listening In

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In 1890, as the telephone’s influence spread across the United States, Judge Robert S. Taylor of Fort Wayne, Ind., told an audience of inventors that the telephone had introduced an “epoch of neighborship without propinquity.” Scientific American called it “nothing less than a new organization of society.” The New York Times reported that two Providence men “were recently experimenting with a telephone, the wire of which was stretched over the roofs of innumerable buildings, and was estimated to be fully four miles in length”:

They relate that on the first evening of their telephonic dissipation, they heard men and women singing songs and eloquent clergymen preaching ponderous sermons, and that they detected several persons in the act of practising on brass instruments. This sort of thing was repeated every evening, while on Sunday morning a perfect deluge of partially conglomerated sermons rolled in upon them. … The remarks of thousands of midnight cats were borne to their listening ears; the confidential conversations of hundreds of husbands and wives were whispered through the treacherous telephone. … The two astonished telephone experimenters learned enough of the secrets of the leading families of Providence to render it a hazardous matter for any resident of that city to hereafter accept a nomination for any office.

In 1897 one London writer wrote, “We shall soon be nothing but transparent heaps of jelly to each other.”

(From Carolyn Marvin, When Old Technologies Were New, 1988.)

Podcast Episode 109: Trapped in a Cave

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Image: Wikimedia Commons

In 1925, Kentucky caver Floyd Collins was exploring a new tunnel when a falling rock caught his foot, trapping him 55 feet underground. In this week’s episode of the Futility Closet podcast we’ll follow the desperate efforts to free Collins, whose plight became one of the first popular media sensations of the 20th century.

We’ll also learn how Ronald Reagan invented a baseball record and puzzle over a fatal breakfast.

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Self Study

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Here’s an isosceles triangle. Sides AB and AC are equal, and this means that the angles opposite those sides are equal as well.

That’s intuitively reasonable, but proving it is tricky. Teachers of Euclid’s Elements came to call it the pons asinorum, or bridge of donkeys, because it was the first challenge that separated quick students from slow.

The simplest proof, attributed to Pappus of Alexandria, requires no additional construction at all. We know that two triangles are congruent if two sides and the included angle of one triangle are congruent to their corresponding parts in the other (the “side-angle-side” postulate). So Pappus suggested simply picking up the triangle above, flipping it over, and putting it down again, to produce a second triangle ACB. Now if we compare the respective parts of the two triangles, we find that angle A is equal to itself, AB = AC, and AC = AB. Thus the “left-hand” angles of the two triangles are congruent, and it follows that the base angles of the original triangle above are equal.

In his 1879 book Euclid and his Modern Rivals, Lewis Carroll accepts the proof but remarks that it “reminds one a little too vividly of the man who walked down his own throat.”

Podcast Episode 105: Surviving on Seawater

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In 1952, French physician Alain Bombard set out to cross the Atlantic on an inflatable raft to prove his theory that a shipwreck victim can stay alive on a diet of seawater, fish, and plankton. In this week’s episode of the Futility Closet podcast we’ll set out with Bombard on his perilous attempt to test his theory.

We’ll also admire some wobbly pedestrians and puzzle over a luckless burglar.

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Podcast Episode 103: Legislating Pi

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In 1897, confused physician Edward J. Goodwin submitted a bill to the Indiana General Assembly declaring that he’d squared the circle — a mathematical feat that was known to be impossible. In today’s show we’ll examine the Indiana pi bill, its colorful and eccentric sponsor, and its celebrated course through a bewildered legislature and into mathematical history.

We’ll also marvel at the confusion wrought by turkeys and puzzle over a perplexing baseball game.

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Podcast Episode 91: Voyage of the Damned

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In 1939, an ocean liner carrying 900 Jewish refugees left Nazi Germany seeking sanctuary in North America, but it was turned away by every nation it appealed to. In this week’s episode of the Futility Closet podcast we’ll follow the so-called “voyage of the damned” and the plight of its increasingly desperate passengers.

We’ll also discuss the employment prospects for hermits in Seattle and puzzle over the contentment of a condemned woman.

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