Self Study

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.”

May 14, 2016May 13, 2016 | Science & Math

Parenthood

On Mother’s Day, May 14, 1939, Cleveland Indians pitcher Bob Feller took his mother to Comiskey Park to see him pitch against the Chicago White Sox. Lena Feller, who had traveled 250 miles from Van Meter, Iowa, with her husband and daughter, sat in the grandstand between home and first base and watch her son amass a 6-0 lead in the first three innings.

Then, in the bottom of the third, Chicago third baseman Marvin Owen hit a line drive into her face.

Feller was following through with his pitching motion and saw it happen. “I felt sick,” he wrote later, “but I saw that Mother was conscious. … I saw the police and ushers leading her out and I had to put down the impulse to run to the stands. Instead, I kept on pitching. I felt giddy and I became wild and couldn’t seem to find the plate. I know the Sox scored three runs, but I’m not sure how.”

The injury was painful but not serious. Feller managed to win the game (9-4) and then hurried to the hospital. In his 1947 autobiography, Strikeout Story, he wrote, “Mother looked up from the hospital bed, her face bruised and both eyes blacked, and she was still able to smile reassuringly. ‘My head aches, Robert,’ she said, ‘but I’m all right. Now don’t go blaming yourself … it wasn’t your fault.'”

May 13, 2016May 11, 2016 | Entertainment · Oddities

A Rolling Lemon

My lousy car has an odometer without 4s — in every position, the counter advances from 3 directly to 5. For example, when it read 000039 I drove one mile and watched it roll over to 000050. Today the odometer reads 002005. How many miles has the car actually traveled?

	Select Click for Answer
Because it’s using 9 digits, the odometer is recording the mileage in base 9, except that its digits 5, 6, 7, 8, and 9 represent the base-9 digits 4, 5, 6, 7, and 8. So the actual (base 10) mileage today is just 2004₍₉₎, or 2 × 9³ + 4 = 2 × 729 + 4 = 1462. (From the 2005 American Mathematics Competition.)

May 12, 2016May 22, 2016 | Puzzles

Wing Men

Bombers in World War I were typically manned by two crew members, a pilot and an observer. The pilot operated the forward machine gun and the observer the rear one, so they depended on one another for their survival. In addition, the two men would share the same hut or tent, eat their meals together, and often spend all their free time together. This closeness produced “some remarkable and amusing results,” writes Hubert Griffith in R.A.F. Occasions (1941):

There were pilots who took the precaution of teaching their observers to fly, with the primitive dual-control fitted to the R.E.8 of those days — and at least one couple who used to take over the controls almost indiscriminately from one another: there was the story that went round the mess, of Creaghan (the pilot) arriving down out of the air one day and accusing his observer of having made a bad landing, and of Vigers, the observer, in turn accusing Creaghan of having made a bad landing. It turned out on investigation that each of them had thought the other to be in control of the aircraft; that because of this neither of them, in fact, had been in control at all; and that, in the absence of any guiding authority, the machine had made a quite fairly creditable landing on her own.

Griffith writes, “It was, I suppose, the most personal relationship that ever existed.”

May 11, 2016May 10, 2016 | History · Oddities · Technology

Left or Right?

bicycle puzzle

You come upon the track of a bicycle in the mud. Was the bicycle traveling to the left or the right?

	Select Click for Answer
On a conventional bicycle the rear wheel is fixed — at any given moment it’s traveling toward the front wheel’s point of contact with the ground. This means that a tangent drawn to the rear wheel’s track will always intersect the front wheel’s track. The reverse is not necessarily true — in the diagram above, the yellow tangent drawn to one curve doesn’t intersect the other curve at all. So the curve with the yellow tangent must correspond to the front wheel. Now, which way was the bicycle going? Choose some points on the rear wheel’s curve and draw tangents at those points, extending them in both directions until they reach the front wheel’s curve. This corresponds to putting the actual bicycle on the diagram, with its rear wheel at the tangent point, and assessing the position the front wheel would take in either direction, left and right. The distance between the wheels is constant, so we want to choose the direction in which the length of the segments is unchanging. Above, the leftward segments of the blue and green tangents are the same length, but the rightward segments are different, so the bicycle must have been traveling from right to left.

Click for Answer

May 10, 2016May 22, 2016 | Puzzles

New Tricks

A British officer in the 44th regiment, who had occasion, when in Paris, to pass one of the bridges across the Seine, had his boots, which had been previously well polished, dirtied by a poodle Dog rubbing against them. He in consequence went to a man who was stationed on the bridge, and had them cleaned. The same circumstance having occurred more than once, his curiosity was excited, and he watched the Dog. He saw him roll himself in the mud of the river, and then watch for a person with well polished boots, against which he contrived to rub himself. Finding that the shoeblack was the owner of the Dog, he taxed him with the artifice; and, after a little hesitation, he confessed that he had taught the Dog the trick in order to procure customers for himself. The officer being much struck with the Dog’s sagacity, purchased him at a high price, and brought him to England. He kept him tied up in London for some time, and then released him. The dog remained with him a day or two, and then made his escape. A fortnight afterwards he was found with his former master, pursuing his old trade on the bridge.

— Samuel Griswold Goodrich, Tales of Animals, 1835

May 9, 2016 | Oddities

Podcast Episode 105: Surviving on Seawater

alain bombard

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.

See full show notes …

May 8, 2016October 10, 2017 | Podcast · Science & Math

Travel Delays

https://commons.wikimedia.org/wiki/File:SSJ100_for_Interjet_-_Interiors_(9016257074).jpg — Image: Wikimedia Commons

What’s the most effective strategy for loading an airplane? Most airlines tend to work from the back to the front, accepting first the passengers who will sit in high-numbered rows (say, rows 25-30), waiting for them to find their seats, and then accepting the next five rows, and so on. Both the airline and the passengers would be glad to know that this is the most effective strategy. Is it?

In 2005, computer scientist Eitan Bachmat of Ben-Gurion University decided to find out. He devised a model that considers parameters of the aircraft cabin, the boarding method, the passengers, and their behavior, and found that the most important variable is a combination of three parameters: the length of the aisle blocked by a standing passenger, multiplied by the number of seats in a row, divided by the distance between rows. If rows are 80 centimeters apart, there are six seats in a row, and a standing passenger and his hand luggage take up 40 centimeters of the aisle, then the passengers headed for a single row will block the aisle space of three rows while they’re waiting to reach their seats.

This quickly backs things up. Even if the airline admits only passengers with row numbers 25-30, half the aisle will be completely blocked and most passengers will have to wait until everyone in front of them has sat down before they reach their seats. The time it takes to fill the cabin grows in proportion to the number of passengers.

A better policy would be to call up the passengers in rows 30, 27, and 24; then those in 29, 26, and 23; and so on (perhaps using color-coded boarding passes). These combinations of passengers would not block one another in the aisles.

An even better policy, Bachmat found, would be to dispense with seat assignments altogether and let passengers board the plane and pick their seats as they please. “With this method, or lack of a method,” writes George Szpiro, “the time required to get people on board and into their seats would only be proportional to the square root of the number of passengers.”

(Eitan Bachmat et al., “Analysis of Airplane Boarding Times,” Operations Research 57:2 [2009]: 499-513 and George S. Szpiro, A Mathematical Medley, 2010. See All Aboard.)

May 8, 2016May 7, 2016 | Science & Math

Persons and Promises

A conundrum by Oxford philosopher Derek Parfit:

In several years, a young Russian will inherit vast estates. Because he has socialist ideals, he intends, now, to give the land to the peasants. But he knows that in time his ideals may fade. To guard against this possibility, he does two things. He first signs a legal document, which will automatically give away the land, and which can be revoked only with his wife’s consent. He then says to his wife, ‘Promise me that, if I ever change my mind, and ask you to revoke this document, you will not consent.’ He adds, ‘I regard my ideals as essential to me. If I lose these ideals, I want you to think that I cease to exist. I want you to regard your husband then, not as me, the man who asks for this promise, but only as his corrupted later self. Promise me that you will not do what he asks.’

She agrees. In time the Russian’s ideals fade, and when he inherits the land he asks his wife to revoke the document, declaring that he releases her from her earlier commitment. What is her obligation here? She had made her promise to her earlier husband, but is that a different person from the man before her now? In her view, the man to whom she made her promise no longer exists, and so cannot release her from her obligation. Is this right? If a person is a succession of earlier and later selves, does a promise attach to a person or to a self?

(Derek Parfit, Reasons and Persons, 1984. See The Ulysses Contract.)

May 7, 2016May 6, 2016 | Oddities

In a Word

syntomy
n. brevity, conciseness

First published in January 1981, NASA Reference Publication 1059, “Space Transportation System and Associated Payloads: Glossary, Acronyms, and Abbreviations,” is a list of “compressed identifiers of systems or structures felt too long and cumbersome to be christened in the normal fashion.”

Among the entries are BX, for box, FLG, for flag, and FLP, for flap.

In Words (1983), Paul Dickson writes, “One is hard-pressed to think of a situation in which an abbreviation that saves only one letter actually saves time and causes less confusion.”

May 6, 2016 | Language