Science & Math

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!

Math Notes

88 + 88 + 58 + 98 + 38 + 48 + 78 + 78 = 88593477

Long Addition" title="2009-10-23-long-addition

In The Hunting of the Snark, the Butcher confirms for the Beaver that Two and One are Three:

Taking Three as the subject to reason about–
A convenient number to state–
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.

The result we proceed to divide, as you see,
By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.

Fittingly for Carroll, the math works:

snark math

Mirror Twins

42263001 is a perfect square, and so is its reversal, 10036224.


If Martians are observing us, how can we show them we’re intelligent?

Carl Friedrich Gauss proposed marking a huge right triangle on the Siberian plain; Austrian astronomer Joseph von Littrow suggested carving a perfect circle in the Sahara and filling it with burning kerosene.

Joseph Pulitzer favored a more direct approach: He wanted to build a huge billboard in New Jersey recommending his newspaper to inquiring Martians.

He pressed the idea until an assistant asked, “What language shall we print it in?”

Perron’s Paradox

Let N be the largest positive integer. Then either N = 1 or N > 1.

If N > 1 then N2 > N, which breaks our definition of N as the largest integer. Therefore N = 1.

“The implications of this paradox are devastating,” writes Laurence Chisholm Young. “In seeking the solution to a problem, we can no longer assume that this solution exists. Yet this assumption has been made from time immemorial, right back in the beginnings of elementary algebra, where problems are solved by starting off with the phrase: ‘Let x be the desired quantity.’”

What’s in a Name

The disciples of Descartes made a perfect anagram upon the Latinised name of their master, ‘Renatus Cartesius,’ one which not only takes up every letter, but which also expresses their opinion of that master’s speciality–’Tu scis res naturae’ (Thou knowest the things of nature).

– William T. Dobson, Poetical Ingenuities and Eccentricities, 1882


A guaranteed way to win at roulette, from Eugene Northrop’s Riddles in Mathematics (1945):

  1. Bet $1 on red.
  2. If you win, go to step 6. If you lose, bet $2 on red.
  3. If you win, go to step 6. If you lose, bet $4 on red.
  4. If you win, go to step 6. If you lose, bet $8 on red.
  5. (And so on.)
  6. When you win, you’ll be $1 ahead. Go back to step 1.

“Theoretically, of course, it is possible for the bank to wipe you out financially. Actually, however, runs of more than 10 or 12 successive blacks or reds are extremely rare, and your stake at the twelfth play would be only $2048. When you do win you will, as before, be $1 ahead of the bank. You can then begin all over again. Simple, isn’t it?”

Benardete’s Paradox

Prometheus angers Zeus, who dispatches an army of demons with these instructions:

  • Demon 1: If Prometheus is not dead in one hour, kill him.
  • Demon 2: If Prometheus is not dead in half an hour, kill him.
  • Demon 3: If Prometheus is not dead in quarter of an hour, kill him.

And so on. When Prometheus is found dead, the council of gods is displeased, but they find it impossible to identify the guilty demon — any suspect can point to an infinity of demons who must have acted before him. Must Zeus go free?

The Morning Star Paradox

As it happens, the morning star and the evening star are both Venus–but the solar system might have evolved so that Mercury, for instance, was the brightest star in the morning sky.

Thus the morning star has a property that the evening star does not have: It’s necessarily identical with the morning star.

And if the morning star and the evening star have different properties, then they’re not the same object after all.

Thinking Back

Can you move an object using only your mind? Of course not. But can you move one in the past?

Since January 1997, the Retropsychokinesis Project at the University of Kent has invited Web visitors to try to influence the replay of a prerecorded bitstream. In other words, they must try to influence an event that has already happened.

The experimenters claim to be agnostic as to whether retroactive causality exists, but “the best existing database suggests that the odds are in the order of 1 in 630 thousand million that the experimental evidence is the result of chance.”

Try it for yourself here — but remember, if you have some skepticism about this, it may only be because someone in the future is influencing you.

The Paradox of Omnipresence and Timelessness

It’s an essential attribute of God that he’s omnipresent, and Thomas Aquinas held that he also stands somehow outside of time and is not bound by temporal considerations. But, Richard La Croix argues,

if God is indeed omnipresent then it would appear that he must have been in the United Nations building yesterday as well as the day before yesterday. And if God was in the United Nations building both yesterday and the day before, then it would appear that he is in time and that temporal predicates do apply to him. So, it would appear that God is not a timeless being if he is omnipresent and that two doctrines crucial to the theology of Thomas Aquinas are logically incompatible.

Omniscience poses further problems: If God knows all things, then he knows what both man and he himself will do. So how is free will possible?

The Wow! Signal

On Aug. 15, 1977, a telescope at Ohio State University detected a strong narrowband radio signal in the constellation Sagittarius — one so unusual that astronomer Jerry Ehman marked the printout with an exclamation.

The signal’s intensity rose and then fell as the beam swept past its position in the sky. That’s consistent with an extraterrestrial origin … but in 30 years and more than 100 searches, no one has been able to relocate it.

Without a recurrence, there’s no way to know what Ehman’s telescope heard that night — it’s just a frustrating splash in a large, silent sea.

Hard Times

In 1820, Richard Whatley wrote a facetious elegy for Oxford geologist William Buckland:

Where shall we our great Professor inter,
That in peace he may rest his bones?
If we hew him a rocky sepulchre
He will rise and break the stones,
And examine each stratum that lies around
For he’s quite in his element underground.

Ironically, when Buckland did pass away in 1856, the gravedigger struck an outcrop of limestone just below the surface and had to use gunpowder to put Buckland to rest.

Ambrose Bierce defined geology as “The science of the earth’s crust–to which, doubtless, will be added that of its interior whenever a man shall come up garrulous out of a well.”

Zalcman’s Paradox

Is the question “What is an example of a question which is not its own answer?” its own answer?

The Paradox of the Knower

No one knows that this sentence is true.

That sentence can’t be false, because that would lead immediately to a contradiction.

But if it’s true, then omniscience is impossible.

Therefore there can be no all-knowing being.

The Ulam Spiral

Write the numbers from 41 to 440 in a square spiral:

ulam spiral

Remarkably, all the numbers on the red diagonal are prime — even when the spiral is continued into a 20 × 20 square.

No one’s quite sure what to make of this. Polish mathematician Stanislaw Ulam discovered the pattern while doodling at a scientific meeting in 1963.

The Self-Flowing Flask'sSelfFlowingFlask.png

Denis Papin described this perpetual motion scheme in 1685. The pint of liquid in the goblet weighs more than the ounce in the tube, and so forces it down. When the liquid spills out the top of the tube and back into the goblet, it starts a cycle that must continue until the liquid finally evaporates. Where is the error?

Pieces of Pi

In the decimal expansion of π:

  • the digits 27182818 — the first eight digits of e — appear at position 1,526,800.
  • the digits 14142135 — the first eight digits of the square root of 2 — appear at position 52,638.
  • the first eight digits of π itself — 31415926 — reappear at position 50,366,472.
  • 16470 appears at position 16470.
  • there are seven 7s at position 3,346,228, eight 8s at position 46,663,520, and six 9s at position 762.
  • The White House switchboard number (456-1414) is at position 3,193,808, the population of France (65,073,482) is at position 98,709,092, and Disneyland’s zip code (92802) is at position 41,112.

Write out the alphabet starting with J:


Erase all letters that have left-right symmetry (such as A) and count the letters in each of the five groups that remain. (James Davis)

Sum Caws

The Russian for crow (a bird) in the genitive case plural is sorok. The same word also means forty. Hence, the ambiguous construction ’100 crows + 100 crows = 200 crows’ can also mean ’140 + 140 = 280.’

– V.M. Bradis, Lapses in Mathematical Reasoning, 1938