“A Kiss and Its Consequences,” English carte de visite, 1910.

In 1965, Caltech computer scientist Donald Knuth privately circulated a theorem that, “under special circumstances, 1 + 1 = 3”:

Proof. Consider the appearance of John Martin Knuth, who exhibits 
the following characteristics:

Weight      8 lb. 10 oz.      (3912.23419125 grams)         (3)
Height      21.5 inches          (0.5461 meters)            (4)
Voice          loud               (60 decibels)             (5)
Hair         dark brown       (Munsell 5.0Y2.0/11.8)        (6)


He conjectured that the stronger result 1 + 1 = 4 might also be true, and that further research on the problem was contemplated. “I wish to thank my wife Jill, who worked continuously on this project for nine months. We also thank Dr. James Caillouette, who helped to deliver the final result.”

(From Donald E. Knuth, Selected Papers on Fun & Games, 2011.)


A Scrabble player needs a way to recognize the potential in any collection of tiles. If your rack contains the seven letters AIMNSTU, for example, what eighth letter should you be watching for to create an acceptable eight-letter word?

If you arrange your seven letters into the word TSUNAMI, and if you’ve memorized the corresponding phrase COASTAL HARM, then you have your answer: Any of the letters in that phrase will produce an acceptable eight-letter word:


TSUNAMI: COASTAL HARM is an example of an anamonic (“anagram mnemonic”), a tool that tournament players use to memorize valuable letter combinations. Devising useful anamonics is itself an art form in the Scrabble community — one has to create a memorable phrase using a constrained set of letters. Some are memorable indeed:


“One of the first anamonics I ever read, back in 1998, was PRIEST: EVERYONE COMPLAINED OF THE SODOMY,” wrote Jeff Myers in Word Ways in May 2007. “I couldn’t believe it. The letters in that phrase — no more and no less — could combine with PRIEST to make 7-letter words.”

When the word list TWL06 appeared, PERITUS became a legal word. That’s PRIEST + U, so the mnemonic phrase now needed to include a U. “One simple fix is: EVERYONE COMPLAINED OF YOUTH SODOMY,” wrote Myers. “Now maybe even more startling.”

John Chew maintains canonical lists of anamonics using the official Tournament Word List and the alternate SOWPODS list.

Podcast Episode 116: Notes and Queries

In this week’s episode of the Futility Closet podcast we’ll explore some curiosities and unanswered questions from Greg’s research, including the love affair that inspired the Rolls Royce hood ornament, a long-distance dancer, Otto von Bismarck’s dogs, and a craftily plotted Spanish prison break.

We’ll also run after James Earl Ray and puzzle over an unsociable jockey.


Workers constructing Washington’s Grand Coulee Dam in 1942 fed a cable through a 500-foot drain by tying a string to an alley cat’s tail.

A 2001 earthquake in Olympia, Wash., drew a graceful rose with a sand-tracing pendulum.

Sources for this week’s feature:

The best source I can find regarding the origins of the Rolls Royce hood ornament is this Telegraph article from 2008, in which Montagu’s son says, “My father and Eleanor shared a great passion. It was a grand love affair – perhaps even the love of his life. All this happened before my father met my mother. But I understand my father’s first wife knew about the mistress. She was very tolerant of her and they got on very well.” But this quote is given in the service of promoting a film about the affair, which makes it less objective than I’d like. (Paul Tritton of the Rolls-Royce Owners’ Club of Australia disputes the story here.)

Alexandre Dumas’ habit of eating an apple every morning beneath the Arc de Triomphe is described in this New York Times article, among many other modern sources. The earliest mention I can find is a 1911 article in the Dietetic and Hygienic Gazette, attributing the intervention to Hungarian physician David Gruby. I’ve confirmed that Gruby served as a physician to Dumas (père et fils), but I can’t find anything about an apple.

The incidents of the Savoy Hotel cloakroom and the Travellers Club suicide are both described in N.T.P. Murphy’s A Wodehouse Handbook (2013). The suicide rule is mentioned at the end of this Telegraph article, which gives me hope that it’s true, but I can’t find anything more comprehensive.

The story of the Providence United Methodist Church is told in both Randy Cerveny’s Freaks of the Storm (2005) and Rick Schwartz’s Hurricanes and the Middle Atlantic States (2007). Snopes says it’s “mostly true.”

In Constable’s Clouds, published by the National Galleries of Scotland 2000, Edward Morris writes, “It is this moment of early morning light — and what has been described as ‘the atmosphere of stillness tinged with expectancy’ — that Constable translates into the finished canvas.”

Judith Collins mentions Joseph Beuys’ responsibility for snow in her introduction to Andy Goldsworthy’s Midsummer Snowballs (2001).

Reader Olga Izakson found the description of Tiras, Otto von Bismarck’s “dog of the empire,” in Robert K. Massie’s Dreadnought (1991). A few further links.

The role of Esperanto in the planning of the 1938 San Cristobál prison break is described (I think) here.

In 1600 William Kemp published a pamphlet chronicling his 1599 morris dance to Norwich, Kemps Nine Daies Wonder, to quiet doubters.

The allegation that Margaret Thatcher ordered the identities of British government employees to be encoded in the word spacing of their documents appears in Gregory Kipper’s Investigator’s Guide to Steganography (2003). I’ve found it in other technical documents, but these tend to cite one another rather than an authoritative source.

Listener mail:

Madison Kahn, “60 Hours of Hell: The Story of the Barkley Marathons,” Outside, May 8, 2013.

Wikipedia, “Barkley Marathons” (accessed Aug. 6, 2016).

Wikipedia, “Kaihogyo,” (accessed Aug. 6, 2016).

Adharanand Finn, “What I Learned When I Met the Monk Who Ran 1,000 Marathons,” Guardian, March 31, 2015.

Associated Press, “Japanese Monks Endure With a Vow of Patience,” June 10, 2007.

Here’s a corroborating link for this week’s lateral thinking puzzle (warning: spoiler).

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 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 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!

More Morals

la rochefoucauld

Maxims of François VI, Duc de La Rochefoucauld (1613–1680):

  • “We commonly slander more thro’ Vanity than Malice.”
  • “We have more Laziness in our Minds than in our Bodies.”
  • “There are few People but what are ashamed of their Amours when the Fit is over.”
  • “We should not judge of a Man’s Merit by his great Qualities, but by the Use he makes of them.”
  • “He who is pleased with Nobody, is much more unhappy than he with whom Nobody is pleased.”
  • “There are some disguised Falsehoods so like Truths, that ‘twould be to judge ill not to be deceived by them.”
  • “Men sometimes think they hate Flattery, but they hate only the Manner of Flattering.”
  • “Acquired Honor is Surety for more.”
  • “Innocence don’t find near so much Protection as Guilt.”
  • “‘Tis our own Vanity that makes the Vanity of others intolerable.”
  • “‘Tis a common Fault to be never satisfied with ones Fortune, nor dissatisfied with ones Understanding.”
  • “Envy is more irreconcilable than Hatred.”
  • “‘Tis better to employ our Understanding, in bearing the Misfortunes that do befall us, than in foreseeing those that may.”
  • “A good Head finds less Trouble in submitting to a wrong Head than in conducting it.”
  • “Folly attends us close thro’ our whole Lives; and if anyone seems wise, ’tis merely because his Follies are proportionate to his Age, and Fortune.”

And “As ’tis the Characteristic of a great Genius to say much in a few Words, small Geniuses have on the contrary the Gift of speaking much and saying nothing.”

Moessner’s Theorem

moessner's theorem

Write out the positive integers in a row and underline every fifth number. Now ignore the underlined numbers and record the partial sums of the other numbers in a second row, placing each sum directly beneath the last entry that it contains.

Now, in this second row, underline and ignore every fourth number, and record the partial sums in a third row. Keep this up and the entries in the fifth row will turn out to be the perfect fifth powers 15, 25, 35, 45, 55

If we’d started by ignoring every fourth number in the original row, we’d have ended up with perfect fourth powers. In fact,

For every positive integer k > 1, if every kth number is ignored in row 1, every (k – 1)th number in row 2, and, in general, every (k + 1 – i)th number in row i, then the kth row of partial sums will turn out to be just the perfect kth powers 1k, 2k, 3k

This was discovered in 1951 by Alfred Moessner, a giant of recreational mathematics who published many such curiosa in Scripta Mathematica between 1932 and 1957.

(Ross Honsberger, More Mathematical Morsels, 1991.)


In March 1945 the Japanese painted the giant image of an American B-29 on the Tien Ho airfield in China. They gave it a burning engine and a 300-foot wingspan, so that when viewed from a great altitude it would look like a stricken bomber flying at several thousand feet. Their hope was that this would induce high-flying Allied planes to drop down to investigate, bringing them within range of their anti-aircraft guns. I don’t know whether it worked.

The Atlantic has a collection of similar deceptive exploits from World War II.

All Roads

Another puzzle from Kendall and Thomas’ Mathematical Puzzles for the Connoisseur (1971):

Take three consecutive positive integers and cube them. Add up the digits in each of the three results, and add again until you’ve reached a single digit for each of the three numbers. For example:

463 = 97336; 9 + 7 + 3 + 3 + 6 = 28; 2 + 8 = 10; 1 + 0 = 1
473 = 103823; 1 + 0 + 3 + 8 + 2 + 3 = 17; 1 + 7 = 8
483 = 110592; 1 + 1 + 0 + 5 + 9 + 2 = 18; 1 + 8 = 9

Putting the three digits in ascending order will always give the result 189. Why?

Click for Answer

In a Word

adj. high-flying

adj. capable of being seen, visible

n. a source of fear

John Lithgow’s eyes pop out of his head momentarily at the climax of “Nightmare at 20,000 Feet,” the final segment in Twilight Zone: The Movie (1983). In the segment, a remake of the famous television episode from 1963, Lithgow plays a nervous air passenger who discovers a gremlin on the wing of his plane. At the moment when he lifts the shade, the edit shows the monster for 17 frames, then Lithgow’s face for 10 frames, then the monster for 42 frames, and then a 5-frame shot of Lithgow’s head incorporating the eye-popping effect.

Of these 5 frames, the first three show a wild-eyed Lithgow, the fourth shows bulging eyes, and the fifth is shown below. “This 5-frame sequence is on the screen for 1/5 second, but the most distorted image is only visible for 1/24 second,” writes William Poundstone in Bigger Secrets. “Blink at the wrong time, and you miss it. But if you watch the shot carefully at normal speed, the sequence is detectable. Lithgow’s eyes seem to inflate with an accelerated, cartoon-like quality.”

Here’s the frame:

twilight zone movie