The presence of an inconvenient roadway prevented architect Donato Bramante from making Milan’s Santa Maria presso San Satiro as grand as he’d hoped — so he painted a fictional apse in a shallow niche that’s only a few feet deep.
Created in 1477, it’s one of the earliest examples of trompe l’oeil in the history of art.
You have 12 coins that appear identical. Eleven have the same weight, but one is either heavier or lighter than the others. How can you identify it, and determine whether it’s heavy or light, in just three weighings in a balance scale?
This is a classic puzzle, but in 1992 Washington State University mathematician Calvin T. Long found a solution “that appears little short of magic.” Number the coins 1 to 12 and make three weighings:
First weighing: 1 3 5 7 vs. 2 4 6 8
Second weighing: 1 6 8 11 vs. 2 7 9 10
Third weighing: 2 3 8 12 vs. 5 6 9 11
To solve the problem, note the result of each weighing and assemble a three-digit numeral in base 3 as follows:
Left pan sinks: 2
Right pan sinks: 0
Balance: 1
For example, if coin 7 is light, that produces the number 021 in base 3. Now converting that to base 10 gives 7, the number of the odd coin, and an examination of the weighings shows that it must be light. Another example: If coin 2 is heavy, then we get 002 in base 3, which is 2 in base 10. Note that it’s possible to get an answer that’s higher than 12, e.g. when coin 7 is heavy — in that case subtract the base-10 answer you get from 26.
Another curious method to solve the classic puzzle, this one involving verbal mnemonics, appeared in Eureka in 1950.
(Calvin T. Long, “Magic in Base 3,” Mathematical Gazette 76:477 [November 1992], 371-376.)
09/30/2018 UPDATE: Due to an error in the original paper, the weighings I originally specified don’t work in every case — in the third weighing, the left pan should contain 2 3 8 12, not 1 2 8 12. I’ve amended this in the post above; everything should work now. Sorry for the error; thanks to everyone who wrote in.
I saw a woman sleeping. In her sleep she dreamt Life stood before her, and held in each hand a gift — in the one Love, in the other Freedom. And she said to the woman, ‘Choose!’
And the woman waited long: and she said: ‘Freedom!’
And Life said, ‘Thou hast well chosen. If thou hadst said, ‘Love,’ I would have given thee that thou didst ask for; and I would have gone from thee, and returned to thee no more. Now, the day will come when I shall return. In that day I shall bear both gifts in one hand.’
I heard the woman laugh in her sleep.
— Olive Schreiner, Dreams, 1891
A neat observation by Japanese mathematician Jun Maekawa: If an origami model can be flattened without damage, then at any vertex (meeting of edges) in its crease pattern the number of “valley” folds and “mountain” folds always differ by two.
The single-vertex crease pattern above has five mountain folds (folds whose outer surface is colored) and three valley folds (folds whose inner surface is colored). (The fifth mountain fold is a bit hard to notice in this example — it’s folded flat at bottom right.)
One consequence of this is that every vertex has an even number of creases, and therefore that the regions between the creases can be colored with two colors.
Paper folder Toshikazu Kawasaki found a related theorem: At any vertex, the sum of all the odd angles is 180 degrees (and likewise the even).
It was at the place afterwards called Solenhofen. The weather was miserable, as Jurassic weather usually was. The rain beat steadily down, and carbon dioxide was still upon the earth.
The Archaeopteryx was feeling pretty gloomy, for at that morning’s meeting of the Amalgamated Association of Enaliosaurians he had been blackballed. He was looked down upon by the Pterodactyl and the Ichthyosaurus deigned not to notice him. Cast out by the Reptilia, and Aves not being thought of, he became a wanderer upon the face of the earth. ‘Alas!’ sighed the poor Archaeopteryx, ‘this world is no place for me.’ And he laid him down and died; and became imbedded in the rock.
And ages afterward a featherless biped, called man, dug him up, and marvelled at him, crying, ‘Lo, the original Avis and fountain-head of all our feathered flocks!’ And they placed him with great reverence in a case, and his name became a by-word in the land. But the Archaeopteryx knew it not. And the descendant for whom he had suffered and died strutted proudly about the barn-yard, crowing lustily cock-a-doodle-do!
— Samuel P. Carrick Jr., in The Fly Leaf, January 1896
From Lee Sallows:
(Thanks, Lee!)
“Everybody is somebody’s bore.” — Edith Sitwell
Discourage muggers during your evening dog walks with this “werewolf-style” muzzle, from Russia’s Zveryatam pet supply.
I wonder what other dogs make of this.
A soul once cowered in a gray waste, and a mighty shape came by. Then the soul cried out for help, saying, ‘Shall I be left to perish alone in this desert of Unsatisfied Desires?’
‘But you are mistaken,’ the shape replied; ‘this is the land of Gratified Longings. And, moreover, you are not alone, for the country is full of people; but whoever tarries here grows blind.’
— Edith Wharton, The Valley of Childish Things, and Other Emblems, 1896
Given a group of 10 men and 10 women, all straight, is it always possible to pair them off in stable marriages, that is, to pair them so that there exist no man and woman who would prefer each other to the partners they have? Yes:
This process continues until everyone is engaged (as they must be, since every man must eventually propose to every woman and every woman must accept someone). All the marriages are stable because no man can end up pining for a woman who would prefer him to her own partner — that woman must already have rejected or jilted him at some point during the courting:
In 1962, mathematicians David Gale and Lloyd Shapley showed that stable marriages can always be found for any equal number of men and women.
