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.)

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).

“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

or

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.)

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.)

The 2018 Best Illusion of the Year Contest results have been announced, and Kokichi Sugihara has won with a “triply ambiguous object”:

Sugihara is a regular prizewinner — in 2010 he defied gravity.

In 2011, Michael I. Norton of Harvard Business School, Daniel Mochon of Yale, and Dan Ariely of Duke found that test subjects were willing to pay 63% more for IKEA furniture that they had assembled than for identical units that came preassembled. In a separate study, they found that subjects who had finished building items were willing to pay more for their creations than subjects who had only partially completed assembly. The lesson seems to be that consumers place a disproportionately high value on products they’ve had a hand in creating.

The principle had been understood, though not named, as early as the 1950s, when homemakers initially disdained instant cake mixes, which they felt made cooking too easy and inspired no investment in the outcome. When the recipe was adjusted to require the cook to crack an egg, sales went up.

John Conway’s author biography in Melvin Fitting and Brian Rayman’s 2017 book Raymond Smullyan on Self Reference:

John H. Conway is the John von Neumann Distinguished Professor Emeritus of Applied and Computational Mathematics at Princeton. He has four doctorate degrees: PhD (Cambridge), DSc (Liverpool), D.h.c. (Iasi), PhD (Bremen). He has written or co-written 11 books, one of which has appeared in 11 languages. He has both daughters and sons, each daughter has as many sisters as brothers and each son twice as many sisters as brothers. He has met Raymond Smullyan repeatedly at many Gatherings for Gardner and Andrew Buchanan in Cambridge, New York and Princeton.

The size of his family is left as an exercise.