Matched Set

It’s well known that the sum of the cubes of the first n integers equals the square of their sum:

13 + 23 + 33 + 43 + 53 = (1 + 2 + 3 + 4 + 5)2

California State University mathematician David Pagni found another case in which the sum of cubes equals the square of a sum. Take any whole number:


List all its divisors:

1, 2, 4, 7, 14, 28

Count the number of divisors of each of these:

1 has 1 divisor
2 has 2 divisors
4 has 3 divisors
7 has 2 divisors
14 has 4 divisors
28 has 6 divisors

Now cube these numbers and sum the cubes:

13 + 23 + 33 + 23 + 43 + 63 = 324

And sum the same set of numbers and square the sum:

(1 + 2 + 3 + 2 + 4 + 6)2 = 324

The two results are the same: The sum of the cubes of these numbers will always equal the square of their sum.

(David Pagni, “An Interesting Number Fact,” Mathematical Gazette 82:494 [July 1998], 271-273.)

03/10/2017 UPDATE: Reader Kurt Bachtold points out that this was originally discovered by Joseph Liouville, a fact that I should have recalled, as I’d written about it in 2011. (Thanks, Kurt.)

Shame and Fortune

In 1818 caricaturist George Cruikshank saw several people hanging from a gibbet near Newgate Prison in London and learned to his horror that they had been executed for passing forged one-pound notes — at the time, doing so even unknowingly was punishable by death or transportation.

The fact that a poor woman could be put to death for such a minor offence had a great effect upon me — and I at that moment determined, if possible, to put a stop to this shocking destruction of life for merely obtaining a few shillings by fraud; and well knowing the habits of the low class of society in London, I felt quite sure that in very many cases the rascals who forged the notes induced these poor ignorant women to go into the gin-shops to ‘get something to drink,’ and thus pass the notes, and hand them the change.

He went home and dashed off this sketch, which was then printed on the post paper used by the bank, so that it would resemble counterfeit currency. “The general effect was of a counterfeit, but closer examination revealed that every element of the official design had been replaced by a savage parody,” writes Robert L. Patten in George Cruikshank’s Life, Times, and Art. The seal shows Britannia eating her children, the stamp depicts 12 tiny heads in prison, and the pound sign is a coiled hangman’s rope.

The protest created a sensation, and remedial legislation was passed. Cruikshank’s satire, noted the Examiner, “ought to make the hearts of the Bank Directors ache at the sight.”

Warming Up,_from_A_Memoir_of_Jane_Austen_(1870).jpg

Chapter 1 of Jack and Alice, a novel written by Jane Austen when she was 13 years old:

Mr. Johnson was once upon a time about 53; in a twelvemonth afterwards he was 54, which so much delighted him that he was determined to celebrate his next Birth day by giving a Masquerade to his Children and Freinds. Accordingly on the Day he attained his 55th year tickets were dispatched to all his Neighbours to that purpose. His acquaintance indeed in that part of the World were not very numerous as they consisted only of Lady Williams, Mr and Mrs Jones, Charles Adams and the 3 Miss Simpsons, who composed the neighbourhood of Pammydiddle and formed the Masquerade.

Before I proceed to give an account of the Evening, it will be proper to describe to my reader, the persons and Characters of the party introduced to his acquaintance.

Mr and Mrs Jones were both rather tall and very passionate, but were in other respects, good tempered, wellbehaved People. Charles Adams was an amiable, accomplished and bewitching young Man; of so dazzling a Beauty that none but Eagles could look him in the Face.

Miss Simpson was pleasing in her person, in ther Manners and in her Disposition; an unbounded ambition was her only fault. Her second sister Sukey was Envious, Spitefull and Malicious. Her person was short, fat and disagreeable. Cecilia (the youngest) was perfectly handsome but too affected to be pleasing.

In Lady Williams every virtue met. She was a widow with a handsome Jointure and the remains of a very handsome face. Tho’ Benevolent and Candid, she was Generous and sincere; Tho’ Pious and Good, she was Religious and amiable, and Tho’ Elegant and Agreable, she was Polished and Entertaining.

The Johnsons were a family of Love, and though a little addicted to the Bottle and the Dice, had many good Qualities.

Such was the party assembled in the elegant Drawing Room of Johnson Court, amongst which the pleasing figure of a Sultana was the most remarkable of the female Masks. Of the Males a Mask representing the Sun, was the most universally admired. The Beams that darted from his Eyes were like those of that glorious Luminary tho’ infinitely Superior. So strong were they that no one dared venture within half a mile of them; he had therefore the best part of the Room to himself, its size not amounting to more than 3 quarters of a mile in length & half a one in breadth. The Gentleman at last finding the feirceness of his beams to be very inconvenient to the concourse by obliging them to croud together in one corner of the room, half shut his eyes by which means, the Company discovered him to be Charles Adams in his plain green Coat, without any mask at all.

When their astonishment was a little subsided their attention was attracted by 2 Domino’s who advanced in a horrible Passion; they were both very tall, but seemed in other respects to have many good qualities. ‘These said the witty Charles, these are Mr and Mrs Jones.’ and so indeed they were.

No one could imagine who was the Sultana! Till at length on her addressing a beautifull Flora who was reclining in a studied attitude on a couch, with ‘Oh Cecilia, I wish I was really what I pretend to be’, she was discovered by the never failing genius of Charles Adams, to be the elegant but ambitious Caroline Simpson, and the person to whom she addressed herself, he rightly imagined to be her lovely but affected sister Cecilia.

The Company now advanced to a Gaming Table where sat 3 Dominos (each with a bottle in their hand) deeply engaged, but a female in the character of Virtue fled with hasty footsteps from the shocking scene, whilst a little fat woman representing Envy, sate alternately on the foreheads of the 3 Gamesters. Charles Adams was still as bright as ever; he soon discovered the party at play to be the 3 Johnsons, Envy to be Sukey Simpson and Virtue to be Lady Williams.

The Masks were then all removed and the Company retired to another room, to partake of elegant and well managed Entertainment, after which the bottle being pretty briskly pushed about by the 3 Johnsons, the whole party not excepting even Virtue were carried home, Dead Drunk.

“Jane Austen was born before those bonds which (we are told) protected women from truth, were burst by the Brontës or elaborately untied by George Eliot,” wrote G.K. Chesterton. “Yet the fact remains that Jane Austen knew more about men than either of them. Jane Austen may have been protected from truth: but it was precious little of truth that was protected from her.”

A Snow Dam

flateyri dam 1

The Icelandic fishing village of Flateyri was devastated when an avalanche buried 17 homes in 1995. To guard against further trouble, they built an earthen dam in the shape of an enormous A.

It worked: An avalanche struck the dam’s eastern wing in February 1999, and another struck the western wing the following March. Both were deflected harmlessly into the sea.

flateyri dam 2

Good Luck,_Temperance_Map,_1838_Cornell_CUL_PJM_1049_01.jpg

C. Wiltberger created this allegorical map of temperance in 1838 (click to enlarge). The goal is to get from the Ocean of Animal Appetite in the west to the Ocean of Eternity in the east. It would be natural enough to investigate Indulgence and Generosity Islands, but this will lead you to Evil Company Island, and once you’re through the Devil’s Trap you’ll have to negotiate the Sea of Intemperance, with its islands of Murder, Arson, Larceny, and Quarrel. Beyond the Great Gulf of Wretchedness lies the Sea of Anguish, which puts you out at Suicide Island (and its capital, Spontaneous Combustion).

The better plan is to head north immediately and enter the Cold Water River at Hope Island. Bear south at Knowledge toward Cultureville and Mount Science and take the Tee Total Rail Road to the Sea of Temperance, and then head north through the Old Age Outlet past Comfortville and Restburg and safely into Eternity. (Beware the Gulf of Broken Pledges — even at this late stage, it will lead you directly to Desperation Point.)

My favorite part: Poverty Island has a port called Nosupper.

Podcast Episode 144: The Murder Castle

When detectives explored the Chicago hotel owned by insurance fraudster H.H. Holmes in 1894, they found a nightmarish warren of blind passageways, trapdoors, hidden chutes, and asphyxiation chambers in which Holmes had killed dozens or perhaps even hundreds of victims. In this week’s episode of the Futility Closet podcast we’ll follow the career of America’s first documented serial killer, who headlines called “a fiend in human shape.”

We’ll also gape at some fireworks explosions and puzzle over an intransigent insurance company.


In 1908 a Strand reader discovered an old London horse omnibus on the outskirts of Calgary.

If Henry Jenkins truly lived to 169, then as an English subject he’d have changed religions eight times.

Sources for our feature on H.H. Holmes:

Erik Larson, The Devil in the White City, 2004.

John Borowski, The Strange Case of Dr. H.H. Holmes, 2005.

Harold Schechter, Depraved: The Definitive True Story of H.H. Holmes, 1994.

Alan Glenn, “A Double Dose of the Macabre,” Michigan Today, Oct. 22, 2013.

John Bartlow Martin, “The Master of the Murder Castle,” Harper’s, December 1943.

Corey Dahl, “H.H. Holmes: The Original Client From Hell,” Life Insurance Selling, October 2013.

“Claims an Alibi: Holmes Says the Murders Were Committed by a Friend,” New York Times, July 17, 1895.

“Holmes in Great Demand: Will Be Tried Where the Best Case Can Be Made,” New York Times, July 24, 1895.

“Accused of Ten Murders: The List of Holmes’s Supposed Victims Grows Daily,” New York Times, July 26, 1895.

“The Holmes Case,” New York Times, July 28, 1895.

“Expect to Hang Holmes: Chicago Police Authorities Say They Can Prove Murder,” New York Times, July 30, 1895.

“Chicago and Holmes,” New York Times, July 31, 1895.

“No Case Against Holmes: Chicago Police Baffled in the Attempt to Prove Murder,” New York Times, Aug. 2, 1895.

“Did Holmes Kill Pitzel: The Theory of Murder Gaining Ground Steadily,” New York Times, Nov. 20, 1894.

“Holmes Fears Hatch: Denies All the Charges of Murder Thus Far Made Against Him,” New York Times, Aug. 2, 1895.

“Quinlan’s Testimony Against Holmes: They Think He Committed Most of the Murders in the Castle,” New York Times, Aug. 4, 1895.

“Modern Bluebeard: H.H. Holmes’ Castles Reveals His True Character,” Chicago Tribune, Aug. 18, 1895.

“The Case Opened: A Strong Plea, by the Prisoner for a Postponement,” New York Times, Oct. 29, 1895.

“Holmes and His Crimes: Charged with Arson, Bigamy, and Numerous Murders,” New York Times, Oct. 29, 1895.

“Holmes Grows Nervous: Unable to Face the Portrait of One of His Supposed Victims,” New York Times, Oct. 30, 1895.

“Holmes Is Found Guilty: The Jury Reaches Its Verdict on the First Ballot,” New York Times, Nov. 3, 1895.

“Holmes Sentenced to Die: The Murderer of Benjamin F. Pietzel to Be Hanged,” New York Times, Dec. 1, 1895.

“The Law’s Delays,” New York Times, Feb. 4, 1896.

“Holmes’ Victims,” Aurora [Ill.] Daily Express, April 13, 1896.

“Holmes Cool to the End,” New York Times, May 8, 1896.

Rebecca Kerns, Tiffany Lewis, and Caitlin McClure of Radford University’s Department of Psychology have compiled an extensive profile of Holmes and his crimes (PDF).

Listener mail:

The Seest disaster:

Wikipedia, “Seest Fireworks Disaster” (accessed March 3, 2017).

“Dutch Fireworks Disaster,” BBC News, May 14, 2000.

Wikipedia, “Enschede Fireworks Disaster” (accessed March 3, 2017).

“Vuurwerkramp,” Visit Enschede (accessed March 3, 2017).

Beverly Jenkins, “10 Worst Fireworks Disasters Ever,” Oddee, July 4, 2013.

Jessie Guy-Ryan, “Inside the World’s Deadliest Fireworks Accident,” Atlas Obscura, July 4, 2016.

Wikipedia, “Puttingal Temple Fire” (accessed March 3, 2017).

Rajiv G, “Kollam Temple Fire: Death Toll Reaches 111, 40 Badly Wounded,” Times of India, April 12, 2016.

This week’s lateral thinking puzzle was contributed by listener Daniel Sterman, who sent this corroborating link (warning — this spoils the puzzle).

You can listen using the player above, download this episode directly, or subscribe on iTunes or Google Play Music 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 we’ve set up some rewards to help thank you for your support.

You can also make a one-time donation on the Support Us page 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 Music Animation Machine

Berkeley software engineer Stephen Malinowski creates animated graphical scores of musical works.

“The vertical positions of the bars on the screen represent the relative pitches, while the color can represent instruments or voices, thematic material or tonality,” explains Crétien van Campen in The Hidden Sense. “When they are synchronized, the sound and image are easily linked in our perception. Musical structures like Bach’s canons or his many-voiced compositions thus become understood and accessible by means of a visual aid.”

There’s much more on Malinowski’s YouTube channel; here are some of his favorites.

For the Record

At one point in Samuel Beckett’s 1951 novel Molloy, the title character finds himself at the seaside and “lays in a store of sucking stones”:

They were pebbles but I call them stones. Yes, on this occasion I laid in a considerable store. I distributed them equally between my four pockets, and sucked them turn and turn about. This raised a problem which I first solved in the following way. I had say sixteen stones, four in each of my four pockets these being the two pockets of my trousers and the two pockets of my greatcoat. Taking a stone from the right pocket of my greatcoat, and putting it in my mouth, I replaced it in the right pocket of my greatcoat by a stone from the right pocket of my trousers, which I replaced by a stone from the left pocket of my trousers, which I replaced by a stone from the left pocket of my greatcoat, which I replaced by the stone which was in my mouth, as soon as I had finished sucking it. Thus there were still four stones in each of my four pockets, but not quite the same stones.

It occurs to him that this method won’t ensure that every stone is eventually sucked, and he works out a plan that will achieve this. This takes eight pages, “one of the longest and most detailed accounts of someone working at a mathematical problem in a work of fiction,” according to Richard Phillips in Numbers: Facts, Figures and Fiction.

Maddeningly, in the end Molloy throws away all the stones but one, “for they all tasted exactly the same.”

Banishing Gloom

quin historical atlas

For his Historical Atlas of 1830, Edward Quin took a different approach than other cartographers: Rather than present history as a series of discrete moments, he illustrates the growth of knowledge by covering the earth in obscuring clouds that are beaten back from panel to panel.

“In Quin’s Historical Atlas, the world is shown first in darkness, with clouds obscuring everything outside the Garden of Eden,” note Anthony Grafton and Daniel Rosenberg in Cartographies of Time. “Gradually, as history reveals more of the world, the clouds roll back. Turning the pages of the atlas is a bit like riffling through a flip book, watching darkness recede and the world known to Europeans grow.”