A Hidden Door

Image: Wikimedia Commons

Choose any four points on a circle and join them to form a quadrilateral. Drawing the diagonals of this quadrilateral produces four overlapping triangles (each diagonal creates two of them).

Draw the largest possible circle in each of these triangles, and the centers of these circles will always form a rectangle:

The Ben Franklin Effect


In his autobiography, Benjamin Franklin describes mollifying a rival legislator in the Pennsylvania statehouse:

Having heard that he had in his library a certain very scarce and curious book, I wrote a note to him, expressing my desire of perusing that book, and requesting he would do me the favour of lending it to me for a few days. He sent it immediately, and I return’d it in about a week with another note, expressing strongly my sense of the favour. When we next met in the House, he spoke to me (which he had never done before), and with great civility; and he ever after manifested a readiness to serve me on all occasions, so that we became great friends, and our friendship continued to his death.

This seems to be a real psychological phenomenon — you can sometimes more reliably make a friend by asking a favor than by doing one, or, as Franklin put it, “He that has once done you a kindness will be more ready to do you another, than he whom you yourself have obliged.”

In a 1969 study, subjects who had won money in a question-and-answer competition were asked to return it; those whom the researcher himself approached reported liking him more than those who’d been approached by a secretary. In another study, students were assigned a teaching task using two different methods, one in which they encouraged their students and one in which they insulted and criticized them. In a debriefing they rated the students they’d encouraged to be more likable and attractive than those they’d insulted. That may reveal a converse principle, that we devalue others in order to justify wronging them.

(Jon Jecker and David Landy, “Liking a Person as a Function of Doing Him a Favour,” Human Relations 22:4 [1969], 371-378; John Schopler and John S. Compere, “Effects of Being Kind or Harsh to Another on Liking,” Journal of Personality and Social Psychology 20:2 (1971), 155.)



A quirky old gent, name of Freud,
Was, not without reason, anneud
That his concept of Id,
And all that Id did,
Was so starkly and loosely empleud.

— Martin Fagg

“If you dream,” said the eminent Freud,
“Your Id is in doubt, or annoyed,
By neuroses complex
From suppression of sex,
So passions are best if enjoyed.”

— Russell Miller

Sigmund Freud says that one who reflects
Sees that sex has far-reaching effects,
For bottled-up urges
Come out in great surges
In directions that no-one expects.

— Peter Alexander

Said Freud: “I’ve discovered the Id.
Of all your repressions be rid.
It won’t ease the gravity
Of all the depravity,
But you’ll know why you did what you did.”

— Frank Richards

A Dissent

Image: Wikimedia Commons

In the famous “Milgram experiment” at Yale in 1961, an experimenter directed each subject (the “teacher”) to give what she believed were increasingly painful electric shocks to an unseen “learner” (really an actor). Psychologist Stanley Milgram found that a surprisingly high proportion of the subjects would obey the experimenter’s instructions, even over the learner’s shouts and protests, to the point where the learner fell silent.

Milgram wrote, “For the teacher, the situation quickly becomes one of gripping tension. It is not a game for him: conflict is intense. The manifest suffering of the learner presses him to quit: but each time he hesitates to administer a shock, the experimenter orders him to continue. To extricate himself from this plight, the subject must make a clear break with authority.”

As it happened, one participant, Gretchen Brandt, had been a young girl coming of age in Germany during Hitler’s rise to power and repeatedly exposed to Nazi propaganda during her childhood. During Milgram’s experiment, when the learner began to complain about a “heart condition,” she asked the experimenter, “Shall I continue?” After administering what she thought was 210 volts, she said, “Well, I’m sorry, I don’t think we should continue.”

Experimenter: The experiment requires that you go on until he has learned all the word pairs correctly.

Brandt: He has a heart condition, I’m sorry. He told you that before.

Experimenter: The shocks may be painful but they’re not dangerous.

Brandt: Well, I’m sorry. I think when shocks continue like this they are dangerous. You ask him if he wants to get out. It’s his free will.

Experimenter: It is absolutely essential that we continue.

Brandt: I’d like you to ask him. We came here of our free will. If he wants to continue I’ll go ahead. He told you he had a heart condition. I’m sorry. I don’t want to be responsible for anything happening to him. I wouldn’t like it for me either.

Experimenter: You have no other choice.

Brandt: I think we are here on our own free will. I don’t want to be responsible if anything happens to him. Please understand that.

She refused to continue, and the experiment ended. Milgram wrote, “The woman’s straightforward, courteous behavior in the experiment, lack of tension, and total control of her own action seem to make disobedience a simple and rational deed. Her behavior is the very embodiment of what I envisioned would be true for almost all subjects.”

Asked afterward how her experience as a youth might have influenced her, Brandt said slowly, “Perhaps we have seen too much pain.”

(From Thomas Heinzen and Wind Goodfriend, Case Studies in Social Psychology, 2019.)

Fruit Cocktail


Image: Sridhar Ramesh

This innocent-looking poser has been floating around social media. Trial and error might lead you to the solution (-1,4,11) — that’s not quite valid, as one of the values is negative, but it’s simple enough to be encouraging. Right?

It turns out that the problem is stupendously hard — solving it requires transforming the equation into an elliptic curve, and the smallest positive whole values that work are 80 digits long!

Scottish mathematician Allan MacLeod introduced the problem in 2014, and it found its way onto the web in this Reddit thread. Alon Amit runs through a solution here, but it’s very steep. He writes, “Roughly 99.999995% of the people don’t stand a chance at solving it, and that includes a good number of mathematicians at leading universities who just don’t happen to be number theorists. It is solvable, yes, but it’s really, genuinely hard.”

(Thanks, Chris.)

Podcast Episode 222: The Year Without a Summer

Image: Wikimedia Commons

The eruption of Mount Tambora in 1815 was a disaster for the Dutch East Indies, but its astonishing consequences were felt around the world, blocking the sun and bringing cold, famine, and disease to millions of people from China to the United States. In this week’s episode of the Futility Closet podcast we’ll review the volcano’s devastating effects and surprising legacy.

We’ll also appreciate an inverted aircraft and puzzle over a resourceful barber.

See full show notes …

Math Notes

Each of the numbers 102564, 128205, 153846, 179487, 205128, and 230769 quadruples when its last digit is moved to the first position.

And this property is retained when each is concatenated with itself, as many times as desired (102564102564102564 × 4 = 410256410256410256).

A Fool’s Logic

“It is true that you may fool all the people some of the time; you can even fool some of the people all the time; but you can’t fool all of the people all the time.”

This is commonly attributed to Abraham Lincoln, though it’s not clear that he actually said it. In 2004 mathematician Paul Stockmeyer noticed that its meaning is somewhat ambiguous, too. If we use P(x) to denote the predicate “x is a person,” T(y) to denote the predicate “y is a time,” and F(x, y) to denote the two-argument predicate “x is fooled at time y,” then the first phrase of the quotation, “It is true that you may fool all the people some of the time,” might mean either

\displaystyle  \forall x\left ( P\left ( x \right ) \Rightarrow \exists y\left ( T\left ( y \right ) \wedge F\left ( x, y \right )\right )\right )


\displaystyle  \exists y\left ( T\left ( y \right ) \wedge \forall x\left ( P\left ( x \right )\Rightarrow F\left ( x, y \right ) \right )\right ).

The first statement means “For every possible x, if x is a person then there exists a y such that y is a time and moreover x is fooled at time y” (or, more coloquially, “For every person, there is a time when that person is fooled”).

The second means “There exists a y such that y is a time and moreover for every x, if x is a person then x is fooled at time y (or “There is a time when everyone is simultaneously fooled”).

Which is the right interpretation? Stockmeyer polled his classes and found them nearly equally divided. And that’s only the first phrase of the quotation! Does the second phrase, “you can even fool some of the people all the time,” mean that there are people who remain constantly fooled about everything — or that you can always find a fool at any given time?

“However they are interpreted, they serve as a wonderfully effective preparation for his main point contained in the third phrase,” Stockmeyer writes. “And this phrase, with two quantifiers of the same type, is completely unambiguous.”

(Paul K. Stockmeyer, “What Did Lincoln Really Mean?” College Mathematics Journal 35:2 [2004], 103-104.)



Navigators from the Poluwat atoll of Micronesia find their way among islands by appealing to parallax — a reference island at one side of their course will appear to pass beneath a succession of stars:

The star bearings of the reference island from both the starting and ending points of the trip are known, since on another occasion the reference island may itself become a destination. In between there are other navigation star positions under which the reference island will pass as it ‘moves’ backwards. Its passage under each of these stars marks the end of one etak and the beginning of another. Thus the number of star positions which lie between the bearing of the reference island as seen from the island of origin and its bearing as seen from the island of destination determine the number of etak, which can here be called segments, into which the voyage is conceptually divided. When the navigator envisions in his mind’s eye that the reference island is passing under a particular star he notes that a certain number of segments have completed and a certain proportion of the voyage has therefore been accomplished.

This is a dynamic model: Where Western navigators think of a vessel moving among stationary islands, the Poluwatese find it more natural to think of the canoe as stationary and the islands as moving around it. “Etak is perfectly adapted for its use by navigators who have no instruments, charts, or even a dry place in which to spread a chart if they had one,” writes Stephen D. Thomas in The Last Navigator. “Etak allows the Micronesian navigator to process all his information — course, speed, current drift, and so on — through a single, sea-level perspective.”

(Thomas Gladwin, East Is a Big Bird: Navigation and Logic on Puluwat Atoll, 1970, quoted in Lorenzo Magnani, Philosophy and Geometry, 2001.)

A Modest Proposal

Image: Flickr

While a visiting fellow at All Souls College, Oxford, in 1978, Claude Shannon pondered a personal challenge he faced there:

An American driving in England is confronted with a wild and dangerous world. The cars have the driver on the right and he is supposed to drive on the left side of the road. It is as though English driving is a left-handed version of the right-handed American system.

I can personally attest to the seriousness of this problem. Recently my wife and I, together with another couple on an extended visit to England, decided to jointly rent a car. … With our long-ingrained driving habits the world seemed totally mad. Cars, bicycles and pedestrians would dart out from nowhere and we would always be looking in the wrong direction. The car was usually filled with curses from the men and with screams and hysterical laughter from the women as we careened from one narrow escape to another.

His solution was “grandiose and utterly impractical — the idle dream of a mathematician”:

How will we do this? In a word, with mirrors. If you hold your right hand in front of a mirror, the image appears as a left hand. If you view it in a second mirror, after two reflections it appears now as a right hand, and after three reflections again as a left hand, and so on.

Our general plan is to encompass our American driver with mirror systems which reflect his view of England an odd number of times. Thus he sees the world about him not as it is but as it would be after a l80° fourth-dimensional rotation.

A corresponding adjustment to the steering system will turn the car left when the driver steers right, and vice versa. And filling the cabin with a high-density liquid will reverse the feeling of centrifugal force as well. “A snorkel provides for his breathing and altogether, with our various devices, he feels very much as though he were at home in America!”

(Claude E. Shannon, “The Fourth-Dimensional Twist, or a Modest Proposal in Aid of the American Driver in England,” typescript, All Souls College, Oxford, Trinity term, 1978; via Jimmy Soni and Rob Goodman, A Mind at Play: How Claude Shannon Invented the Information Age, 2017.)