A Curious Equality


The area of the largest equilateral triangle that fits inside a unit square exactly equals the side of the largest square that fits inside a unit equilateral triangle.

This proof is due to John Conway. The largest square that fits inside a unit equilateral triangle is shown at the top; the side of the square is 2\sqrt{3}-3. The largest equilateral triangle that fits inside a unit square is one that’s canted at a 15° angle, shown at the bottom.

Now, the area of the shaded parallelogram is just its base times its height, or 2\sqrt{3}-3. And the parallelogram can be dissected into three pieces that fit precisely into the canted triangle. So the side of the square at the top equals the area of the triangle at the bottom.


“All my life I wanted to be somebody, but now I see I should have been more specific.” — Lily Tomlin

Using One’s Head


Do animals reason? British psychologist C. Lloyd Morgan offered the case of his own dog:

Tony, the fox-terrier, already introduced to my readers, when he wants to go out into the road, puts his head under the latch of the gate, lifts it, and waits for the gate to swing open. Now an observer of the dog’s intelligent action might well suppose that he clearly perceived how the end in view was to be gained, and the most appropriate means for effecting his purpose. The following chain of ideas might be supposed to pass through the dog’s mind, not, indeed, in a clear-cut logical form, but at any rate in a rough and practically serviceable way: ‘Why does that gate remain shut? The latch holds it. I’ll lift the latch. Now it is no longer held, therefore it swings open.’ But is it necessary to assume that there were ideas involving, even in the most rudimentary way, the why and the wherefore? May not the action be quite well explained on the hypothesis that the dog acted under the sole guidance of sense-experience?

Two observers might debate that question for some time, Morgan wrote, one arguing that Tony’s feat must be based on experience and the other insisting that it showed an understanding of how the gate operates. But Morgan himself had had the benefit of seeing the trick evolve. “I was sitting at a window above the garden, and heard the dog put out of the door. I therefore watched him. He ran up and down the low wall, and put his head out between the iron bars, now here, now there, now elsewhere, anxiously gazing into the road. This he did for quite three or four minutes. At length it so happened that he put out his head beneath the latch, which, as I have said, is at a convenient height for his doing so, being about a foot above the level of the wall. The latch was thus lifted. He withdrew his head, and began to look out elsewhere, when he found that the gate was swinging open, and out he bolted.”

So Tony’s trick was the product of a fortunate accident, not abstract reasoning. Morgan proposed a general rule: “In no case is an animal activity to be interpreted in terms of higher psychological processes if it can be fairly interpreted in terms of processes which stand lower in the scale of psychological evolution and development.” This has become known as Morgan’s canon.

(From Morgan’s 1903 book An Introduction to Comparative Psychology.)



The Viking 1 orbiter brought some surprised attention to the Cydonia region of Mars in 1976 when its cameras discovered what appeared to be an enigmatic face staring up into the heavens.

The “face on Mars” has since been explained as an optical illusion, but it recalls a project conceived 30 years earlier by the American artist Isamu Noguchi. Sculpture to Be Seen From Mars, below, was proposed as a massive earthwork to be constructed in “some unwanted area,” perhaps a desert, at an enormous scale, so that the nose would be 1 mile long. When seen from space, the face would show that a civilized life form had once existed on Earth. Noguchi had been embittered by his experiences as a Japanese-American during World War II and the development of atomic weapons; he had originally called the piece Memorial to Man.

noguchi, sculpture to be seen from mars

If those two don’t have enough to talk about, there’s a newcomer to join them: In 2013, face recognition software discovered the image below in a photo of the moon’s south pole taken by NASA’s Lunar Reconnaissance Orbiter. Who’s next?


Podcast Episode 71: Godless in Missouri

In 1880, freethinking attorney George Walser tried a new experiment in the American heartland — a community dedicated against Christianity, “the only town of its size in the world without a priest, preacher, saloon, God or hell.” In this week’s episode of the Futility Closet podcast, we’ll tell the story of Liberal, Missouri — its founding, its confrontations with its Christian neighbors, and its ironic downfall.

We’ll also puzzle over how a woman can suddenly be 120 miles away in just a few minutes.

liberal plat

Sources for our feature on Liberal, Mo.:

J.P. Moore, This Strange Town — Liberal, Missouri: A History of the Early Years, 1880 to 1910, 1963.

Lawrence O. Christensen, William E. Foley, and Gary Kremer, Dictionary of Missouri Biography, 1999.

Tom Flynn, ed., The New Encyclopedia of Unbelief, 2007.

Steve Everly, “History of Southwest Missouri Town Shows Triumph of Faith Over Skepticism,” Nevada Daily Mail, Dec. 26, 2001.

Marvin Vangilder, “Missouri Town Might Assure Stockton as Atheist Target,” Associated Press, Sept. 4, 1963.

“Necrology,” Missouri Historical Review, July 1910.

“Missouri Geography: Community Experiments,” in Walter Barlow Stevens, Missouri the Center State: 1821-1915, Volume 2, 1915.

George Henry Walser, The Life and Teachings of Jesus, 1909.

This week’s lateral thinking puzzle was contributed by listener David White, who sent these corroborating links (warning — these spoil the puzzle).

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!



Russian science writer Yakov Perelman asks: How many times must an 8-toothed cogwheel rotate on its axis to circle around a 24-toothed cogwheel?

Click for Answer

“A Large Turtle”

Captain Augustus G. Hall and the crew of the schooner Annie L. Hall vouch for the following: On March 30, while on the Grand Bank, in latitude 40° 10′, longitude 33°, they discovered an immense live trunk turtle, which was at first thought to be a vessel bottom up. The schooner passed within twenty-five feet of the monster, and those on board had ample opportunity to estimate its dimensions by a comparison with the length of the schooner. The turtle was at least 40 feet long, 30 feet wide, and 30 feet from the apex of the back to the bottom of the under shell. The flippers were 20 feet long. It was not deemed advisable to attempt its capture.

Scientific American, May 12, 1883

Black and White

tanner chess problem

By John Tanner. White to mate in two moves.

Click for Answer

Man Handling


In 1946, sociologist Mirra Komarovsky asked American undergraduate women whether they had “played dumb” on dates. Some of their responses:

  • I am engaged to a southern boy who doesn’t think too much of the woman’s intellect. In spite of myself, I play up to his theories because the less one knows and does, the more he does for you and thinks you “cute” into the bargain. … I allow him to explain things to me in great detail and to treat me as a child in financial matters.
  • When my date said that he considers Ravel’s Bolero the greatest piece of music ever written, I changed the subject because I knew I would talk down to him.
  • One of the nicest techniques is to spell long words incorrectly once in a while. My boyfriend seems to get a great kick out of it and writes back, “Honey, you certainly don’t know how to spell.”
  • Once I went sailing with a man who so obviously enjoyed the role of a protector that I told him I didn’t know how to sail. As it turned out he didn’t either. We got into a tough spot, and I was torn between a desire to get a hold of the boat and a fear to reveal that I had lied to him.
  • I am better in math than my fiancé. But while I let him explain politics to me, we never talk about math even though, being a math major, I could tell him some interesting things.
  • I was once at a work camp. The girls did the same work as the boys. If some girls worked better, the boys resented it fiercely. The director told one capable girl to slow down to keep peace in the group.
  • On dates I always go through the “I-don’t-care-anything-you-want-to-do” routine. It gets monotonous but boys fear girls who make decisions. They think such girls would make nagging wives.
  • I am a natural leader and, when in the company of girls, usually take the lead. That is why I am so active in college activities. But I know that men fear bossy women, and I always have to watch myself on dates not to assume the “executive” role. Once a boy walking to the theater with me took the wrong street. I knew a short cut but kept quiet.

In all, 60 percent said they had “concealed some academic honor, pretended ignorance of some subject, or allowed the man the last word in an intellectual discussion.” “And the funny part of it is that the man, I think, is not always so unsuspecting,” one said. “He may sense the truth and become uneasy in the relation. ‘Where do I stand? Is she laughing up her sleeve or did she mean this praise? Was she really impressed with that little speech of mine or did she only pretend to know nothing about politics?’ And once or twice I felt that the joke was on me: the boy saw through my wiles and felt contempt for me for stooping to such tricks.”

(Mirra Komarovsky, “Cultural Contradictions and Sex Roles,” American Journal of Sociology 52:3 [November 1946], 184-189.)

In a Word


adj. abusive, foul-mouthed, reviling

In his Recollections of the Civil War, Charles Anderson Dana called Union general Andrew Atkinson Humphreys “one of the loudest swearers that I ever knew.” “The men of distinguished and brilliant profanity in the war were General Sherman and General Humphreys — I could not mention any others that could be classed with them. General Logan also was a strong swearer, but he was not a West Pointer: he was a civilian. Sherman and Humphreys would swear to make everything blue when some dispatch had not been delivered correctly or they were provoked.”

In Rex v. Sparling (1722), a leather dresser named James Sparling was alleged in the course of 10 days to “profanely swear fifty-four oaths, and profanely curse one hundred and sixty curses, contra formam statuti.” His conviction was overturned because the charge sheet had failed to list them. “For what is a profane oath or curse is a matter of law, and ought not to be left to the judgment of the witness … it is a matter of great dispute among the learned, what are oaths and what curses.”

When in 1985 a man named Callahan called a California highway patrolman a “fucking asshole,” California Court of Appeal Justice Gerald Brown referred to this phrase as the “Callahan epithet” to avoid having to repeat it continuously, “which arguably would assist its passage into parlor parlance.” And he reversed Callahan’s conviction:

A land as diverse as ours must expect and tolerate an infinite variety of expression. What is vulgar to one may be lyric to another. Some people spew four-letter words as their common speech such as to devalue its currency; their repetition dulls the senses; Billingsgate thus becomes commonplace. Not everyone can be a Daniel Webster, a William Jennings Bryan or a Joseph A. Ball. …

Fifty years ago the words ‘damn’ and ‘hell’ were as shocking to the sensibilities of some people as the Callahan epithet is to others today. The first word in Callahan’s epithet has many meanings. When speaking about coitus, not everyone can be an F.E. Smith (later Earl of Birkenhead) who, in his speech in 1920 in the House of Commons on the Matrimonial Causes Act, referred to ‘that bond by which nature in its ingenious telepathy has contrived to secure and render agreeable the perpetuation of the species.’

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