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

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 Thanks for listening!

The Hidden Psalm

The final movement on John Coltrane’s 1965 album A Love Supreme is a “musical narration” of a devotional poem that Coltrane included in the album’s liner notes — he put the handwritten poem on a music stand and “played” it as if it were music.

“Coltrane’s hushed delivery sounds deliberately speechlike,” write Ashley Kahn in his 2003 history of the album. “He hangs on to the ends of phrases, repeats them as if for emphasis. He is in fact ‘reading’ through his horn.”

The hidden psalm was marked by New York musicians for decades before Rutgers University musicologist Lewis Porter presented a formal analysis to the American Musicological Society in 1980. “You will find that he plays right to the final ‘Amen’ and then finishes,” he writes in his 1997 biography of the saxophonist. “There are no extra notes up to that point. You will have to make a few adjustments in the poem, however: Near the beginning where it reads, ‘Help us resolve our fears and weaknesses,’ he skips the next line, goes on to ‘In you all things are possible,’ then plays ‘Thank you God’ … towards the end he leaves out ‘I have seen God.'”

“I think music can make the world better and, if I’m qualified, I want to do it,” Coltrane had said. “I’d like to point out to people the divine in a musical language that transcends words. I want to speak to their souls.”

(Thanks, Jeff.)

Math and Poetry

In 1972 the Belgian mathematician Edouard Zeckendorf established Zeckendorf’s theorem: that every positive integer can be represented as the sum of non-consecutive Fibonacci numbers in one and only one way.

In 1979 French poet Paul Braffort celebrated this with a series of 20 poems, My Hypertropes. Each of the 20 poems in the series is informed by the foregoing poems that make up its Zeckendorff sum. For example, the Zeckendorff representation of 12 is 8 + 3 + 1, so poem 12 in Braffort’s sequence shares some characters or images with each of these poems. This forced Braffort to build scenarios that would permit these relations as he wrote the poems.

Each of the numbers 1, 2, 3, 5, 8, and 13 is its own Zeckendorff representation, so Braffort related each of these to its two foregoing Fibonacci numbers (e.g., 8 = 3 + 5). This means that only the first poem, “The Preallable Explanation (or The Rhyme’s Reason),” is not influenced by any of the others. Here is that first poem, as translated by Amaranth Borsuk and Gabriela Jaurequi:

This is my work, this is my study,
like Jarry, Cyrano puffy,

to split hairs on Rimbaud
and on willies find booboos.

If it was fair or if it snowed
in Lhassa Emma Sophie Bo-

vary widow of slow carnac
gave herself to the god of wack.

Leibnitz, saying: “Verse …” What an ac-
tor for this superb “Vers …”. Oh “nach”!

He aims, Emma, the apoplexy
of those drunk on galaxy.

At the club of “spinach” kings (nay,
Bach never went there, Banach yea!)

Leibnitz — his graph ibo: not six
mus, three nus, one phi, bona xi —

haunts without profit Bonn: “Ach! Gee
if I were great Fibonacci!!! …”

Now, for example, Poem 12, “MODELS (for Petrovich’s Band),” is an alexandrine with two six-line stanzas. The Zeckendorff representation of 12 is 1 + 3 + 8, so in each stanza of Poem 12 the first line is influenced by Poem 1, the third by Poem 3, and the sixth by Poem 8, each drawing on specific lines in the source poem. The first line in the sixth couplet of Poem 1, “He aims, Emma, the apoplexy,” informs the first line of Poem 12, “For a sweet word from Emma: a word for model”; the second line of the sixth couplet from Poem 1, “of those drunk on galaxy,” informs the first line of the second stanza in Poem 12, “Our galaxies have already packed their valise”; the phrase “when I saw you / weave a letter to Elise” in Poem 3 becomes “they say from this time forth five letters to Elise” in Poem 12; and the couplet “And Muses who compose / They’re a troop they’re tropes” in Poem 8 becomes “Tragic tropes: Leonardo is Fibonacci.”

“Thus, Braffort’s collection of poems, My Hypertropes, has an internal structure provided by a mathematical theorem,” writes Robert Tubbs in Mathematics in Twentieth-Century Literature and Art (2014). “The structure does not entirely determine these poems, but it does provide connections between the poems that might not be there otherwise.”

Knights and Scoundrels

A problem from the 1994 Italian Mathematical Olympiad:

Every inhabitant on the island of knights and scoundrels is either a knight (who always tells the truth) or a scoundrel (who always lies). A visiting journalist interviews each inhabitant exactly once and gets the following answers:

A1: On this island there is at least one scoundrel.
A2: On this island there are at least two scoundrels.

An-1: On this island there are at least n – 1 scoundrels.
An: On this island everyone is a scoundrel.

Can the journalist decide whether the knights outnumber the scoundrels?

Click for Answer


Many thanks, everyone, for all your messages in response to my 10-year anniversary post, in which I asked for your ideas as to how we might keep Futility Closet going. It generated such a huge number of responses that I fear I won’t be able to respond to everyone individually. I’m reading and carefully considering every idea, and then I need to talk to some people in order to refine a plan. I’ll keep you posted as we go along. I really appreciate all your expressions of good will and your contributions, both creative and material. I hope we can find a way forward, if only for the chance to keep writing for such a wonderful audience.

If you’d been meaning to reach out and haven’t yet, please do — I always want to hear from you. You can reach me at

Ersatz English

The lyrics in Italian singer Adriano Celentano’s 1972 single “Prisencolinensinainciusol” sound like American English, but they’re gibberish.

“Ever since I started singing, I was very influenced by American music and everything Americans did,” he told NPR in 2014. “So at a certain point, because I like American slang — which, for a singer, is much easier to sing than Italian — I thought that I would write a song which would only have as its theme the inability to communicate. And to do this, I had to write a song where the lyrics didn’t mean anything.”

It reached number 4 on the Belgian charts in 1973.

Cutting Up
Image: Wikimedia Commons

Choose any number of points on a circle and connect them to form a polygon.

This polygon can be carved into triangles in any number of ways by connecting its vertices.

No matter how this is done, the sum of the radii of the triangles’ inscribed circles is constant.

This is an example of a Sangaku (literally, “mathematical tablet”), a class of geometry theorems that were originally written on wooden tablets and hung as offerings on Buddhist temples and Shinto shrines during Japan’s Edo period (1603-1867). This one dates from about 1800.