The Peters Projection,_date_line_in_Bering_strait.svg

In 1967 German filmmaker Arno Peters promoted a new map of the world in which areas of equal size on the globe appear of equal sizes on the map, so that poor, less powerful nations near the equator are restored to their rightful proportions.

Peters promoted the map by comparing it the popular Mercator projection, which is useful to navigators but makes Europe appear larger than South America and Greenland larger than China.

Peters’ goal was to empower underdeveloped nations, which he felt had suffered from “cartographic imperialism.” But his own map badly distorts the polar regions — cartographic educator Arthur Robinson wrote that its “land masses are somewhat reminiscent of wet, ragged, long winter underwear hung out to dry on the Arctic Circle” — and observers noted that Peters’ native Germany suffered less distortion than the underdeveloped nations he was trying to help.

To quell what they felt was an ill-founded controversy, in 1990 seven North American geographic organizations adopted a resolution urging media and government to stop using all rectangular world maps “for general purposes or artistic displays,” as they necessarily distort the planet’s features. That included both Mercator’s and Peters’ projections.

Peters’ map wasn’t even new. It had first been proposed by Scottish clergyman James Gall — who had noted in 1885 that “we may obtain comparative area with mathematical accuracy” by using this projection, but “we must sacrifice everything else.”

A Call for Change

The most common coins in U.S. circulation are worth 1¢, 5¢, 10¢, and 25¢. University of Waterloo computer scientist Jeffrey Shallit found that with this system the average cost of making change is 4.7; that is, if every amount of change between 0¢ and 99¢ is equally likely to be needed, then on average a change-maker must return 4.7 coins with each transaction.

Can we do better? Shallit found two four-coin sets that reduce the average cost to a minimum: (1¢, 5¢, 18¢, 25¢) and (1¢, 5¢, 18¢, 29¢). Either reduces the average cost to 3.89.

“We would therefore gain about 17% efficiency in change-making by switching to either of these four-coin systems,” he writes. And “the first system, (1, 5, 18, 25), possesses the notable advantage that we only need make one small alteration in the current system: replace the current 10¢ coin with a new 18¢ coin.”

(Jeffrey Shallit, “What This Country Needs Is an 18¢ Piece,” Mathematical Intelligencer 25:2 [June 2003], 20-23.)

Even Sevens

A three-digit number is evenly divisible by 7 if and only if twice its first digit added to the number formed by its two last digits gives a result that’s divisible by 7. So, for example, 938 is divisible by 7 because 2 × 9 + 38 = 56 = 7 × 8.

In fact this can be extended to numbers of any length: 229187 → 2 × 2291 + 87 = 4669 → 2 × 46 + 69 = 161 → 2 × 1 + 61 = 63 = 7 × 9.

(J. Kashangaki, “A Test for Divisibility by Seven,” Mathematical Gazette 80:487 [March 1996], 226.)

Figure and Ground

Because we live on land, we tend to make maps in which oceans are afterthoughts, mere spaces between the continents. In 1986 oceanographer Athelstan Spilhaus sought to remedy this with a cartographic puzzle in which seven pieces can be combined in various ways, each presenting “a different, but equally valid, viewpoint of the features of the earth.”

Above, they’re arranged to show that “the land masses of North and South America, on the one hand, and those of Europe, Asia, and Africa on the other, are maintained in associated groupings, while the vastness of the Pacific Ocean has been set off to the left of the map.”

But the same seven pieces might be rearranged to illustrate the fact that “the Pacific Ocean widely separates Asia from the Americas.”

In 1942 Spilhaus had also devised a world map in which the oceans take the forefront, reminding us that Earth is “a water planet, with a single great ocean covering nearly three-quarters of its surface.”

Volume Control

I just thought this was interesting: In the 1970s a little spate of studies investigated why men and women carry books in stereotypically different ways. A 1976 study in Tennessee found that by junior high school males tended to carry books at their sides, with the arm relatively straight and the hand cupped under the book’s lower edge. Females cradled a book in the arm at the front side of the body, resting on the hip or pelvic bone.

A second study in the same year ruled out some theories: It found that men and women carried books of roughly equal weight, and that both had hand grips strong enough to carry their books in either position. (Also, carrying purses didn’t significantly alter the way women carried their books.)

A University of Washington study two years later replicated the earlier findings but suggested that “women with hips that extend past the comfortable fall line of the arm along the side of the body will not show the side carry typically seen in males.” (“In effect, the hip in females fills the side space that males fill with their books.”)

But a re-examination 15 years later found that the picture was changing: While 90 percent of the men still carried books at their sides, now so did 43-60 percent of the women. So perhaps it’s not correct to speak of these as intrinsically masculine and feminine styles. But that raises another question: “why … men’s carrying behavior is uniform and stable, whereas women’s behavior is more varied and changing.”

(Thomas P. Hanaway and Gordon M. Burghardt, “The Development of Sexually Dimorphic Book-Carrying Behavior,” Bulletin of the Psychonomic Society 7:3 [1976], 267-270; Philip J. Spottswood and Gordon M. Burghardt, “The Effects of Sex, Book Weight, and Grip Strength on Book-Carrying Styles,” Bulletin of the Psychonomic Society 8:2 (1976), 150-152; Judith D. Scheman, Joan S. Lockard, and Bruce L. Mehler, “Influences of Anatomical Differences on Gender-Specific Book-Carrying Behavior,” Bulletin of the Psychonomic Society 11:1 [1978], 17-20; Evelyne Thommen, Emiel Reith, and Christiane Steffen, “Gender-Related Book-Carrying Behavior: A Reexamination,” Perceptual and Motor Skills 76:2 [1993], 355-362.)

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.)