Science & Math

Perpetual Notion

The balls on the right exert greater torque than those on the left, so the wheel ought to turn forever, right?

Sadly, the balls on the left are more numerous.

“If at first you don’t succeed,” wrote Quentin Crisp, “failure may be your style.”

Math Notes

multigrade equation

Multiplication Palindromes

12 × 42 = 24 × 21
12 × 63 = 36 × 21
12 × 84 = 48 × 21
13 × 62 = 26 × 31
23 × 96 = 69 × 32
24 × 63 = 36 × 42
24 × 84 = 48 × 42
26 × 93 = 39 × 62
36 × 84 = 48 × 63
46 × 96 = 69 × 64
14 × 82 = 28 × 41
23 × 64 = 46 × 32
34 × 86 = 68 × 43
13 × 93 = 39 × 31

Immortal Truth

In Scripta Mathematica, March 1955, Pedro A. Pisa offers an unkillably valid equation:

123789 + 561945 + 642864 = 242868 + 323787 + 761943

Hack away at its terms, from either end, and it remains true:

beiler equation math

Stab it in the heart, removing the two center digits from each term, and it still balances:

1289 + 5645 + 6464 = 2468 + 3287 + 7643

Do this again and it still balances:

19 + 55 + 64 = 28 + 37 + 73

Most amazing: You can square every term above, in every equation, and they’ll all remain true.

Math Notes

math notes pyramid

Nature, Nurture

Identical twins Jack Yufe and Oskar Stohr were born in 1932 to a Jewish father and a Catholic mother. Their parents divorced when the boys were six months old; Oskar was raised by his grandmother in Czechoslovakia, where he learned to love Hitler and hate Jews, and Jack was raised in Trinidad by his father, who taught him loyalty to the Jews and hatred of Hitler.

At 47 they were reunited by scientists at the University of Minnesota. Oskar was a conservative who enjoyed leisure, Jack a liberal workaholic. But both read magazines from back to front, both wore tight bathing suits, both wrapped rubber bands around their wrists, both liked sweet liqueur and spicy foods, both had difficulty with math, both flushed the toilet before and after using it — and both enjoyed sneezing suddenly in elevators to startle other passengers.

See Doppelgangers.


To discover whether a number is divisible by 11, add the digits that appear in odd positions (first, third, and so on), and separately add the digits in even positions. If the difference between these two sums is 0 or a multiple of 11, the original number is divisible by 11. Otherwise it’s not.

For example:

11 × 198249381729 = 2180743199019

Sum of digits in odd positions = 2 + 8 + 7 + 3 + 9 + 0 + 9 = 38

Sum of digits in even positions = 1 + 0 + 4 + 1 + 9 + 1 = 16

38 – 16 = 22

22 is a multiple of 11, so 2180743199019 is as well.

Fugitive Truth

When I conduct a psychological experiment, my expectations might influence the outcome.

That’s called the experimenter expectancy effect. Does it exist? Well, we could do an experiment to detect it …

… but if it exists then it would bias the experiment, and if it doesn’t then we’d detect nothing. Either way, it seems, we can’t reliably assess what’s happening.

Warm Work

(Please don’t try this.)

[T]ar … boils at a temperature of 220°, even higher than that of water. Mr. Davenport informs us, that he saw one of the workmen in the King’s Dockyard at Chatham immerse his naked hand in tar of that temperature. He drew up his coat sleeves, dipped in his hand and wrist, bringing out fluid tar, and pouring it off from his hand as from a ladle. The tar remained in complete contact with his skin, and he wiped it off with tow. Convinced that there was no deception in this experiment, Mr. Davenport immersed the entire length of his forefinger in the boiling cauldron, and moved it about a short time before the heat became inconvenient. Mr. Davenport ascribes this singular effect to the slowness with which the tar communicates its heat, which he conceives to arise from the abundant volatile vapour which is evolved ‘carrying off rapidly the caloric in a latent state, and intervening between the tar and the skin, so as to prevent the more rapid communication of heat.’ He conceives also, that when the hand is withdrawn, and the hot tar adhering to it, the rapidity with which this vapour is evolved from the surface exposed to the air cools it immediately. The workmen informed Mr. Davenport, that, if a person put his hand into the cauldron with his glove on, he would be dreadfully burnt, but this extraordinary result was not put to the test of observation.

– David Brewster, Letters on Natural Magic, 1868

“A Very Small Dinner Party”

From Lewis Carroll’s A Tangled Tale: The governor of Kgovjni gives a dinner party for his father’s brother-in-law, his brother’s father-in-law, his father-in-law’s brother, and his brother-in-law’s father — and invites a single person:

Males are denoted by capitals, females by small letters. The governor is E, and his guest is C.

See “Proof That a Man Can Be His Own Grandfather,” No Reunion, and The Half-Bastard.

Math Notes

math notes

Sure Thing

Relativism either applies to itself or it doesn’t.

If it does, then it’s only relatively true.

If it doesn’t, then there’s an absolute truth.

Wheels Within Wheels

If the moon orbits the earth, always presenting the same face to us, does it rotate on its own axis?

It seems a simple question, but its appearance in the London Times in April 1856 set off a war among the English intelligentsia:

  • “A ship sailing round the world presents to the fishes always the same face as the Moon does to us. Coming home again, it will surely not be said that the ship has performed a [rotation].”
  • “Let him perforate a small ivory ball to represent the Moon, pass a wire through it, and bend this wire into a circle of a foot in diameter, and then push the ball round the circumference. Will there then remain any doubt of her not rotating on her axis?”

The answer, as William James would note in his parable of the squirrel, is that “which party is right depends on what you practically mean” by the term in question. Today we’d say that the moon rotates about its axis in the same time it takes to orbit the earth.

Incidentally, Lewis Carroll submitted two letters, but the Times didn’t print them. Perhaps it’s just as well — he was far ahead of everyone else: “I noticed for the first time the fact that though [the moon] only goes 13 times round the earth in the course of the year, it makes 14 revolutions round its own axis, the extra one being due to its motion round the sun.”

The Streetcar Singularity

‘If a man followed the directions of a street-car company,’ said Jones, ‘he would never enter one of its cars. Once in, paradoxically, he would never leave it. Just read that sign; it says, ‘Passengers are forbidden to enter or leave this car while in motion.’ Now, how in the name of Lindley Murray can a passenger do otherwise than get in motion, while leaving or entering a street car?’

— Marshall Brown, Bulls and Blunders, 1893

Math Notes

math notes


A 12th-century version of the liar paradox:

Socrates swears that he will speak only falsehoods about you.

Then he says, “You are a stone.”

This shows that a man can lie and speak the truth at the same time.


  • SCINTILLESCENT contains 7 pairs of letters.
  • Rub two pennies together and you’ll see a third between them.
  • Charles Darwin and Abraham Lincoln were born on the same day.
  • 1285 = (1 + 28) × 5
  • Squeeze an orange peel into a candle flame and you’ll produce a burst of fire.

A Candid Puzzle

Bertrand Russell admired G.E. Moore’s dedication to the truth.

“I have never but once succeeded in making him tell a lie,” he wrote, “and that was by a subterfuge.

“‘Moore,’ I said, ‘do you always speak the truth?’

“‘No,’ he replied.

“I believe this to be the only lie he ever told.”

Stop the World

Arguments against Galileo:

“Animals, which move, have limbs and muscles; the earth has no limbs or muscles, therefore it does not move.” — Scipio Chiaramonti, University of Pisa, 1633

“Buildings and the earth itself would fly off with such a rapid motion that men would have to be provided with claws like cats to enable them to hold fast to the earth’s surface.” — Libertus Fromundus, Anti-Aristarchus, 1631

“If we concede the motion of the earth, why is it that an arrow shot into the air falls back to the same spot, while the earth and all the things on it have in the meantime moved very rapidly toward the east? Who does not see that great confusion would result from this motion?” — Polacco, Anticopernicus Catholicus, 1644

More recent:

“[Astronomers give the rate of Earth's rotation as 1,000 kilometers per hour.] An aircraft flying at this rate in the same direction as that of the rotation could not cover any ground at all. It would remain suspended in mid-air over the spot from which it took off, since both speeds are equal. There would, in addition, be no need to fly from one place to another situated on the same latitude. The aircraft could just rise and wait for the desired country to arrive in the ordinary course of the rotation, and then land; although it is difficult to see how any plane could manage to touch ground at all on an airfield which is slipping away at the rate of 1,000 kilometers per hour. It might certainly be useful to know what people who fly think of the rotation of the earth.” — Gabrielle Henriet, Heaven and Earth, 1957

See No Spin Zone.

Thanks Anyway

In 1776, Viennese schoolmaster Antonio Felkel factored every number up to 408,000. Few people bought the book, though, so the treasury recalled it and used the paper to make ammunition cartridges.

University of Prague professor J.P. Kulik spent 20 years extending the work to 100,000,000. He published it in six volumes in 1867.

Volume 2 has been lost.

The Collatz Conjecture

Think of any whole number greater than zero.

  • If the number is even, divide it by two.
  • If the number is odd, triple it and add one.

If you apply these rules repeatedly, will you always reach 1? Surprisingly, no one knows.

Paul Erdos said, “Mathematics is not yet ready for such confusing, troubling, and hard problems.”

An Artificial Aurora

Karl Selim Lemström worked a quiet miracle in 1882: He strung conducting wire over the summit of a Lapland mountain and watched it draw down a shaft of light from the night sky — poetic proof that the aurora borealis is an electrical discharge from the upper atmosphere.

See Charged Words.

Satanic Compounds

Here’s a sugar alcohol derived from the North Atlantic seaweed Fucus vesiculosus. It’s called fucitol.

And its optical isomers are called D-fuc-ol and L-fuc-ol.

The glycoprotein that vampire bats use to prevent their victims’ blood from clotting is called draculin.

And diethyl azodicarboxylate is explosive, shock-sensitive, carcinogenic, and an eye, skin, and respiratory irritant, which helps to justify its acronym: DEAD.

See Juvenile Chemistry.

Science Marches On

An “infallible remedy against epilepsy,” published in Paris in 1686:

Take of common polypody dried and powdered, of moss growing from the skull of a man who died by violent means (criminals preferred), of nail-filings from human hands and feet, two drachms each; piony root half an ounce, and of fresh misletoe half an ounce. Boil them together as the moon wanes; cool, strain, and administer in small doses.

Cited in Charles White, Three Years in Constantinople, 1846.

See Well, Hey!