The Gingerbread Game

Hansel and Gretl have discovered a gingerbread cottage and are wondering whether to eat some of the tiles on its walls. A witch appears and tells them how they must go about it. “Each of you is to name a whole number between 0 and 100. Hansel’s must be odd and Gretl’s even. No conferring. Whoever chooses the lower number can eat twice that number of gingerbread tiles. Whoever chooses the higher number can eat the lower number.” So, for example, if Hansel chooses 57 and Gretl chooses 30, Hansel will get 30 tiles and Gretl will get 60.

This sounds fine, but the children have just had lessons in game theory and regard this as a non-cooperative game between rational utility maximizers. Gretl knows that Hansel will not choose 99, because 97 would leave him better off if she chose 98 and no worse off if she chose any other number. By the same reasoning, she will avoid 98 and choose 96. In her mind she can follow this train all the way to its end: Rationally, it seems, she must choose 2. Hansel, following it also, finds himself indifferent between 3 and 1. In the end he will receive a paltry two tiles and Gretl either one or four.

Is all of this sound? Gretl says, “There is something radically peculiar about trains of thought which proceed in the subjunctive. You are to work out what you would be rational to do, if I were to choose a number which I shall not choose. I am to do likewise, with each train of thought reproduced inside the other. What happens if either player derails a train by choosing in defiance of it? In that case it becomes radically unclear whether either player still has a rational choice.”

(Martin Hollis, “The Gingerbread Game,” Analysis 54:4 [October 1994], 196-200.)

July 11, 2026July 10, 2026 | Science & Math

Centers of Attraction

In his 1908 autobiography, Francis Galton described a “beauty map” he’d compiled of the British Isles:

Whenever I have occasion to classify the persons I meet into three classes, ‘good, medium, and bad,’ I use a needle mounted as a pricker, wherewith to prick holes, unseen, in a piece of paper. … I used this plan for my beauty data, classifying the girls I passed in streets or elsewhere as attractive, indifferent, or repellent. … I found London to rank highest for beauty; Aberdeen lowest.

In 2008, psychologists Viren Swami and Eliana Hernandez set out to compile a beauty map of their own, this time focusing on London. They asked 461 residents to rate the physical attractiveness of men and women in the city’s 33 boroughs. For the record, the City of London, the City of Westminster, and Kensington and Chelsea were rated highest — which correlates with the affluence but not the health (life expectancy) of the residents in those boroughs.

July 10, 2026July 9, 2026 | Science & Math · Society

A Pressing Appointment

https://commons.wikimedia.org/wiki/File:Kruskal_count_principle.svg — Image: Wikimedia Commons

Choose a number on this clock face and, starting from 12, spell out that number’s English name as you advance clockwise around the face, one letter per numeral. For example, if you’ve chosen 3, count out T-H-R-E-E and you’ll land on the numeral 5. Adopt this new position as your next chosen number and proceed as before (in this case, counting F-I-V-E and landing on 9). After three or more moves you’ll reliably land on 1.

This works because of a characteristic of Markov chains first observed by Russian mathematician Evgenii Borisovich Dynkin. Here’s a card trick that exploits the same principle.

July 8, 2026July 7, 2026 | Science & Math

The Tritone Paradox

This recording presents four pairs of tones, each pair separated by three whole tones, or half an octave. Curiously, some listeners hear the interval as ascending, others as descending. (In fact the tones used are ambiguous as to octave, so there’s no objectively right answer.)

Even more curiously, sometimes a listener’s perception reverses when an interval is transposed, say from C-F# to G#-D, even though nothing else has changed.

Diana Deutsch discovered the effect in 1986.

July 8, 2026July 7, 2026 | Science & Math

Interloper

A pleasing detail from Built for Speed, University of Idaho zoologist John A. Byers’ 2003 account of a year studying pronghorn antelope in western Montana:

“It can be really bizarre if, when I’ve been alone for several hours, I spot a human standing or walking in the distance. For an instant, my reaction is, ‘What the hell is that?’ A large mammal that walks on its hind legs seems very strange.”

July 6, 2026July 5, 2026 | Science & Math

Wiggle Stereoscopy

This street scene, taken in Cork in 1927, seems to give a 3-D effect, but it’s only alternating between two photos taken from slightly different perspectives. The sense of depth comes from parallax and the occlusion of distant objects.

Small animals bob their heads to produce this effect as they plan a jump; it helps them to estimate distance.

June 30, 2026 | Science & Math

Essentials

Albert Einstein and Kurt Gödel used to walk home together from the Institute for Advanced Study in Princeton. In Incompleteness (2005), Rebecca Goldstein gives a sample of their conversation, broached by Gödel:

All of his thinking is governed by an ‘interesting axiom,’ as Ernst Gabor Straus, Einstein’s assistant from 1944 to 1947, once characterized it. For every fact, there exists an explanation as to why that fact is a fact; why it has to be a fact. This conviction amounts to the assertion that there is no brute contingency in the world, no givens that need not have been given. In other words, the world will never, not even once, speak to us in the way that an exasperated parent will speak to her fractious adolescent: ‘Why? I’ll tell you why. Because I said so!’ The world always has an explanation for itself, or as Einstein’s walking partner puts it, Die Welt is vernunftig, the world is intelligible. The conclusions that emanate from the rigorously consistent application of this ‘interesting axiom’ to every subject that crosses the logician’s mind — from the relationship between the body and the soul to global politics to the very local politics of the Institute for Avanced Study itself — often and radically diverge from the opinions of common sense. Such divergence, however, counts as nothing for him. It is as if one of the unwritten laws of his thought processes is: If reasoning and common sense should diverge, then… so much the worse for common sense! What, in the long run, is common sense, other than common?

Somewhat related: Richard Feynman’s sense of “social irresponsibility.”

June 29, 2026June 28, 2026 | Science & Math · Society

Of Thee I Sing

https://commons.wikimedia.org/wiki/File:Set_partitions_4;_Hasse;_circles.svg — Image: Wikimedia Commons

A set of 4 elements can be partitioned in 15 ways.

Pleasingly, this is also the number of rhyme schemes that a 4-line poem can take: AAAA, AAAB, AABA, AABB, AABC, ABAA, ABAB, ABAC, ABBA, ABBB, ABBC, ABCA, ABCB, ABCC, ABCD.

A poem with 5 lines has 52 possible schemes, corresponding to the partitions of a 5-element set, and so on. These are called Bell numbers.

June 25, 2026June 24, 2026 | Science & Math

Gunplay

A chestnut from physics:

Two cannons are aimed directly at one another. One is on the floor of a valley, and the other is on a promontory. Neglecting air resistance, if the two fire simultaneously, what will happen?

	Select Click for Answer
The balls will collide. If there were no gravity, the balls would travel a straight line and meet between the cannons. When gravity is introduced, it imparts the same vertical acceleration to both balls, so they must still meet. Somewhat similar: the monkey and the hunter.

June 22, 2026 | Puzzles · Science & Math

64 Coins

A puzzle from the site Riddle of the Day:

A warden offers a challenge to two prisoners. The first prisoner will enter a room that contains a chessboard. On each of the board’s 64 squares is a coin that’s either heads up or tails up. The guard will identify one square as the “target.”

The first prisoner must turn over exactly one coin and then leave the room. The second prisoner must then enter and, solely by viewing the board, determine which square is the target.

If they succeed, both prisoners will go free. They can confer beforehand on a strategy, but they may not communicate after that. Can they establish a plan that will always work?

	Select Click for Answer
Surprisingly, the answer is yes. Number the squares 0 to 63 and express these numbers in 6-bit binary (000000 to 111111). List the numbers of the squares that bear coins that are heads-up. Below each column in that list, write a 1 if the column contains an odd number of 1s or a 0 if it contains an even number of 1s. For example, suppose that the only heads-up coins are on squares 17, 21, 38, 44, and 55. Listing these in binary and condensing the list as described, we’d get: 010001 010101 100110 101100 110111 ------ 111001 111001 is 57 in decimal. Call that the encoded square; the second prisoner, on just viewing the board, could conduct the same calculations and identify this square independently. Now how can we communicate the identity of the target square? It turns out that, by flipping a single coin, we can move the encoded square to any square we choose. Suppose the target square is square 11. List this below the number of the encoded square, both in binary: 111001 001011 Below each column, write a 0 if the two numbers match and a 1 if they don’t (this is an application of the binary logical operator XOR, “exclusive OR”): 111001 001011 ------ 110010 This result, 110010, is 50 in decimal. This tells us that flipping the coin on square 50 will move the encoded square from square 57 to square 11, the target square. Now we can leave the room, and prisoner 2 can enter and, following the steps above, independently calculate the encoded square, which we’ve now arranged to coincide with the target square. So we both go free. The whole principle is summarized concisely here: (Thanks, Gabriel. Somewhat similar: The Fifth Card.)

Click for Answer

Surprisingly, the answer is yes. Number the squares 0 to 63 and express these numbers in 6-bit binary (000000 to 111111). List the numbers of the squares that bear coins that are heads-up. Below each column in that list, write a 1 if the column contains an odd number of 1s or a 0 if it contains an even number of 1s. For example, suppose that the only heads-up coins are on squares 17, 21, 38, 44, and 55. Listing these in binary and condensing the list as described, we’d get:

111001 is 57 in decimal. Call that the encoded square; the second prisoner, on just viewing the board, could conduct the same calculations and identify this square independently.

Now how can we communicate the identity of the target square? It turns out that, by flipping a single coin, we can move the encoded square to any square we choose.

Suppose the target square is square 11. List this below the number of the encoded square, both in binary:

111001
001011

Below each column, write a 0 if the two numbers match and a 1 if they don’t (this is an application of the binary logical operator XOR, “exclusive OR”):

111001
001011
------
110010

This result, 110010, is 50 in decimal. This tells us that flipping the coin on square 50 will move the encoded square from square 57 to square 11, the target square. Now we can leave the room, and prisoner 2 can enter and, following the steps above, independently calculate the encoded square, which we’ve now arranged to coincide with the target square. So we both go free.

The whole principle is summarized concisely here:

(Thanks, Gabriel. Somewhat similar: The Fifth Card.)

June 21, 2026June 20, 2026 | Science & Math