A Human Cantilever


To illustrate the design principle behind Scotland’s Forth Bridge, engineer Sir Benjamin Baker offered a personal demonstration. Sir John Fowler (left) and Baker (right) each hold two wooden poles with outstretched arms, forming two diamond shapes. When construction foreman Kaichi Watanabe sits in the center, the diamonds are prevented from tipping inward because their outer ends are anchored.

It worked. The bridge, opened in 1890, held the record as the world’s longest single cantilever bridge span for 17 years.

“The All-Purpose Calculus Problem”

kennedy calculus problem

A “calculus problem to end all calculus problems,” by Dan Kennedy, chairman of the math department at the Baylor School, Chattanooga, Tenn., and chair of the AP Calculus Committee:

A particle starts at rest and moves with velocity kennedy integral along a 10-foot ladder, which leans against a trough with a triangular cross-section two feet wide and one foot high. Sand is flowing out of the trough at a constant rate of two cubic feet per hour, forming a conical pile in the middle of a sandbox which has been formed by cutting a square of side x from each corner of an 8″ by 15″ piece of cardboard and folding up the sides. An observer watches the particle from a lighthouse one mile off shore, peering through a window shaped like a rectangle surmounted by a semicircle.

(a) How fast is the tip of the shadow moving?
(b) Find the volume of the solid generated when the trough is rotated about the y-axis.
(c) Justify your answer.
(d) Using the information found in parts (a), (b), and (c) sketch the curve on a pair of coordinate axes.

From Math Horizons, Spring 1994.


“I am a long time in answering your letter, my dear Miss Harriet, but then you must remember that it is an equally long time since I received it — so that makes us even, & nobody to blame on either side.”

— Mark Twain, to an autograph hunter, June 14, 1876

“My room is very easy to find when you get here, and as for distance, you know — why, Oxford is as near to London as London is to Oxford. If your geography-book doesn’t tell you that, it must be a wretched affair.”

— Lewis Carroll, to Mary MacDonald, Jan. 22, 1866

Ice Work


Three hockey pucks, A, B, and C, lie in a plane. You make a move by hitting one puck so that it passes between the other two in a straight line. Is it possible to return all the pucks to their original positions with 1001 moves?

Click for Answer

In a Word


v. to deceive in the manner of a prostitute

BOW-STREET — Eliza Merchant, a black-eyed girl, of that class of women known as ‘unfortunates,’ was charged by Garnet Comerford, a sailor, with robbing him of four sovereigns, several dollars and half-crowns, and his shoes. The tar stated that on Wednesday evening, about eight o’clock he left the house of his Captain, the honourable Mr. Duncan, at the west end of town, intending to pay a visit to a sister, whom he had not seen since he left England in the Seringapatem. On the way, he met as tight a looking frigate as ever he clapt his eyes on. She hoisted friendly colours; he hove to; and they agreed together to steer into port. They sailed up the Strand, when she said she would tow him to a snug berth, and he should share her hammock for the night. He consented; and when he awoke in the morning he found that she had cut and run. His rigging had been thrown all about the room, his four sovereigns and silver, and shoes were carried off.

The Morning Chronicle, Dec. 8, 1828

Fun With Refraction


To show that one can focus sounds waves as well as light waves, Lord Rayleigh would place a ticking pocket watch beyond the earshot of a listener, then introduce a balloon filled with carbon dioxide between them. The balloon acted as a “sound lens” to concentrate the sound, and the listener could hear the watch ticking. Rayleigh would sometimes set the balloon swaying to make the effect intermittent.

Related: Pyrex and Wesson oil have the same index of refraction — so immersing Pyrex in oil makes it disappear:

Curve Stitching

Image: Wikimedia Commons

Mary Everest Boole, the wife of logician George Boole, was an accomplished mathematician in her own right. In order to convey mathematical ideas to young people she invented “curve stitching,” the practice of constructing straight-line envelopes by stitching colored thread through a pattern of holes pricked in cardboard. In each of the examples above, two straight lines are punctuated with holes at equal intervals, defining a quadratic Bézier curve. When the holes are connected with thread as shown, their envelope traces a segment of a parabola.

“Once the fundamental idea of the method has been mastered, anyone interested can construct his own designs,” writes Martyn Cundy in Mathematical Models (1952). “Exact algebraic curves will usually need unequal spacing of the holes and therefore more calculation will be required to produce them; it is surprising, however, what a variety of beautiful figures can be executed which are based on the simple principle of equal spacing.”

The American Mathematical Society has some patterns and resources.

Podcast Episode 41: The Tragic Tale of the Lady Be Good


The American bomber Lady Be Good left North Africa for a bombing run over Italy in 1943. It wasn’t seen again until 15 years later, when explorers discovered its broken remains deep in the Libyan desert. In this episode of the Futility Closet podcast we’ll review the strange history of the lost aircraft and trace the desperate last days of its nine crewmen.

We’ll also climb some twisted family trees and puzzle over the Greek philosopher Thales’ struggles with a recalcitrant mule.

Sources for our segment on the Lady Be Good:

Mario Martinez, Lady’s Men, 1995.

Dennis E. McClendon, The Lady Be Good: Mystery Bomber of World War II, 1962.

Above: The Lady Be Good as she was discovered 440 miles southeast of Benghazi, in remarkably good condition for a plane that had landed itself with one working engine and then lain in the desert for 15 years. The tires on the nose wheel and one of the main landing wheels were undamaged and fully inflated.


The crew: William J. Hatton, pilot; Robert F. Toner, co-pilot; D.P. Hays, navigator; John S. Woravka, bombardier; Harold J. Ripslinger, flight engineer; Robert E. LaMotte, radio operator; Guy E. Shelley Jr., waist gunner; Vernon L. Moore, waist gunner; and S.E. Adams, tail gunner. Hatton, the leader, was probably the first to die. Five months before his posting to Libya, he had written to his mother, “There are about four places they can send me. Arizona, Idaho, and Spokane or Tacoma, Washington. I am sitting here waiting to see which one it is. I hope it isn’t Arizona because I am tired of sand.”

Listener mail:

Our Dec. 21 post “A Man His Own Grandfather,” reprinting an 1868 item about a man whose stepdaughter marries his father, follows a similar post from 2009, “Proof That a Man Can Be His Own Grandfather,” which includes a diagram.

The song “I’m My Own Grandpa” was released by Lonzo & Oscar in 1947. This cover version includes a diagram that explains the relationships:

Thanks to reader David Wright for sending a link to an article in Geneaology Magazine that traces the history of the idea, and to reader Mark Williamson for sharing his own convoluted family tree:

My own mother was an only child, whose father died when she was 9 years old. Her mother then remarried an older man who had several children (and they went on to have several more together). My maternal grandmother’s younger brother was in the military, and when home on leave fell in love with one of my mother’s stepsisters, and they got married and had children of their own. So my grandmother’s brother was my great-uncle, and his wife was my great-aunt, and their children were my second cousins, but he was also my uncle because he was married to my aunt (my mother’s stepsister) and their children were my first cousins. And their father was also their great-uncle, since he was their grandmother’s brother, and therefore their mother was their great-aunt since she was married to their great-uncle. And since they were their great-aunt’s children, that made them their own second cousins.

The first of this week’s two lateral thinking puzzles was inspired by a chance encounter with N.L. Mackenzie’s article “The Nastiness of Mathematicians” in the Pi Mu Epsilon Journal (vol. 9, no. 10, Spring 1994) while toiling at NC State this week. It’s not certain that the story actually befell Thales; the same story is told in Aesop’s fable “The Salt Merchant and His Ass.”

The second puzzle is drawn from Eliot Hearst and John Knott’s excellent 2009 book Blindfold Chess and from Miguel Najdorf’s New York Times obituary (warning: this spoils the puzzle).

Hearst and Knott’s website explains how Najdorf’s longstanding record of 45 blindfold games played simultaneously was broken in 2011 by Marc Lang of Günzburg, Germany. Lang played 46 games and scored +25, =19, -2, as against Najdorf’s astounding +39, =4, -2 in São Paulo in 1947.

You can listen using the player above, download this episode directly, or subscribe on iTunes or via the RSS feed at http://feedpress.me/futilitycloset.

Please consider becoming a patron of Futility Closet — on our Patreon page you can pledge any amount per episode, and all contributions are greatly appreciated. You can change or cancel your pledge at any time, and we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation via the Donate button in the sidebar of the Futility Closet website.

Many thanks to Doug Ross for the music in this episode.

If you have any questions or comments you can reach us at podcast@futilitycloset.com. Thanks for listening!