Suppose that when Marcellus besieged Syracuse, Archimedes was standing at a corner of the city wall. A ditch runs parallel to the wall, separated from it by a distance a. To Archimedes’ left at distance b along the wall stands a catapult, which is distance c from a line perpendicular to the ditch. If Archimedes’ line of sight to the camp runs perpendicular to the wall and the ditch, show that he stood a distance ab/c from the camp.
In 1955, aliens from the planet Clarion contacted a Chicago housewife to warn her that the end of the world was imminent. Psychologist Leon Festinger saw this as a unique opportunity to test a new theory about human cognition. In this week’s episode of the Futility Closet podcast we’ll follow him inside a UFO religion as it approaches the apocalypse.
We’ll also try to determine when exactly LBJ became president and puzzle over some wet streets.
When I heard the learn’d astronomer,
When the proofs, the figures, were ranged in columns before me,
When I was shown the charts and diagrams, to add, divide, and measure them,
When I sitting heard the astronomer where he lectured with much applause in the lecture-room,
How soon unaccountable I became tired and sick,
Till rising and gliding out I wander’d off by myself,
In the mystical moist night-air, and from time to time,
Look’d up in perfect silence at the stars.
That’s Walt Whitman. In 2000, mathematician Mike Keith noted a similar idea in Psalm 19:1-6:
The heavens declare the glory of God;
And the firmament sheweth his handywork.
Day unto day uttereth speech,
And night unto night sheweth knowledge.
There is no speech nor language,
Where their voice is not heard.
Their line is gone out through all the earth,
And their words to the end of the world.
In them hath he set a tabernacle for the sun,
Which is as a bridegroom coming out of his chamber,
And rejoiceth as a strong man to run a race.
His going forth is from the end of the heaven,
And his circuit unto the ends of it:
And there is nothing hid from the heat thereof.
So he married them by rearranging the psalm’s letters:
When I had listened to the erudite astronomer,
When his high thoughts were arranged and charted before me,
When I was shown the length and breadth and height of it,
The Earth, the horned Moon, the chariot of fire,
The hundredth flight of the shuttle through heavyish air,
How soon, mysteriously, I became sad and sick,
Had to wander out, ousted, charging through the forest,
Joining the sure chaos here in a foreign heath,
Having forgotten the vocation of the learned man,
And in the mystic clearing, once more looked up
In perfect silence at the sermon in the stars.
(Michael Keith, “Anagramming the Bible,” Word Ways 33:3 [August 2000], 180-185.)
Hungarian sculptor and architect Ernő Rubik presented this puzzle in 1988; it was originally created by Christopher C. Wiggs and Christopher J. Taylor. The puzzle has two sides, with nine clocks on each side, and the goal is to set all the clocks to 12 o’clock simultaneously.
There are two ways to adjust the clocks. Turning a wheel at any of the four corners will adjust the clock at that corner on both sides of the puzzle. And turning a wheel will also adjust the three clocks adjacent to that corner on one side of the puzzle or the other; which side is determined by the four buttons surrounding the central clock.
So, for example, pressing the northwest button “in” and then turning the northwest wheel will adjust the northwestern quartet of clocks and the corresponding corner clock on the other side of the puzzle. Pulling the northwest button “out” and turning the same wheel will adjust the northwestern clock on the front of the puzzle, its counterpart on the back, and the three clocks adjacent to it on that side.
This is more intuitive than it sounds. Here’s a simulator.
Since there are 14 independent clocks, with 12 settings each, there are a total of 1214 = 1,283,918,464,548,864 possible configurations. It turns out that no configuration requires more than 12 moves to solve; for comparison, in the “worst case” solving a Rubik’s cube can take 20 moves. The trouble, of course, is knowing how to go about it.
My wife and a stranger are both drowning. I can save only one of them. What should I do? In considering this question in 1981, Bernard Williams suggested that I’m having “one thought too many” if I stop to ponder what morality requires; I should save my wife simply because she’s my wife. Troy Jollimore argues that this should silence all other considerations — a husband must “perceive this consideration as possessing such overwhelming importance that it simply drives everything else from his mind.”
But “This sounds terrible,” writes Raja Halwani. “It is one thing to say that Sam should be motivated by the thought that his wife is drowning, but quite another that this thought should silence all others. The danger here is that even if love takes us out of self-absorption, it throws us into the absorption in another, and this does not sound moral.” Is love a moral emotion?
(Bernard Williams, Moral Luck, 1981; Troy Jollimore, Love’s Vision, 2011; Raja Halwani, Philosophy of Love, Sex, and Marriage, 2018.)
The Mad Hatter poses a famous unanswered riddle in Alice’s Adventures in Wonderland: “Why is a raven like a writing desk?”
W.S. Gilbert poses another one in the Gilbert and Sullivan operetta The Yeomen of the Guard. The jester Jack Point asks, “Why is a cook’s brainpan like an overwound clock?” The Lieutenant impatiently says, “A truce to this fooling,” and Jack withdraws, saying, “Just my luck: my best conundrum wasted.”
“Like many in the audience, I have often wondered what the answer to that conundrum is, and one day I put a question about it to Gilbert,” wrote Henry A. Litton in The Secrets of a Savoyard (1922). “With a smile he said he couldn’t tell me then, but he would leave me the answer in his will.”
“I’m sorry to say that it was not found there — maybe because there was really no answer to the riddle, or perhaps because he had forgotten to bequeath to the world this interesting legacy.”
Each number in this pandiagonal order-4 magic square is a three-digit prime:
277 197 631 431 661 401 307 167 137 337 491 571 461 601 107 367
Add 30 to each cell and you get a new magic square, also made up of 16 three-digit primes:
307 227 661 461 691 431 337 197 167 367 521 601 491 631 137 397
Add 1092 to each cell in that one and you get a magic square of four-digit primes:
1399 1319 1753 1553 1783 1523 1429 1289 1259 1459 1613 1693 1583 1723 1229 1489
(Allan William Johnson Jr., “Related Magic Squares,” Journal of Recreational Mathematics 20:1 [January 1988], 26-27, via Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, 2011.)
In 1998, thousands of pigeons mysteriously went missing during two separate races in Virginia and Pennsylvania. More than 2,200 birds vanished, amounting to an 85 percent loss rate. The weather was calm, and it’s normal for a few birds to disappear, but the rate is usually closer to 5 percent.
It’s known that pigeons navigate by the sun and by sensing magnetic fields, but neither of those seems to be the culprit here. “Every year or so, you have one race like this where many disappear,” Cornell zoologist Charles Walcott told the Chicago Tribune. “But what is unusual is to lose so many birds from several races at the same time. What’s going on now is quite mysterious.”
Related: In 2010 a racing pigeon named Houdini disappeared during a 224-mile race in Britain and turned up five weeks later in Panama, 5,200 miles away.
“I was gobsmacked. I didn’t even know where Panama was,” owner Darren Cubberley told the Daily Mirror. “I’ve no idea how Houdini got there — I can only assume she hitched a lift on a ship across the Atlantic.”
The bird, reportedly in “perfect shape,” would have been too expensive to return, so she remained with Gustavo Ortiz, on whose roof she’d landed. At last report she was learning Spanish.
Boston, New York, Philadelphia, Baltimore, Washington D.C., and Atlanta are collinear.
(Thanks, Ryan.)
