Science & Math

Stubborn

Write down any natural number, reverse its digits to form a new number, and add the two:

lychrel number example - 1

In most cases, repeating this procedure eventually yields a palindrome:

lychrel number example - 2
lychrel number example - 3

With 196, perversely, it does not — or, at least, it hasn’t in computer trials, which have repeated the process until it produced numbers 300 million digits long.

Is 196 somehow immune to producing palindromes? No one’s yet offered a conclusive proof — so we don’t know.

Fair Point

One threatening morning as Einstein was about to leave his house in Princeton, Mrs. Einstein advised him to take along a hat.

Einstein, who rarely used a hat, refused.

‘But it might rain!’ cautioned Mrs. Einstein.

‘So?’ replied the mathematician. ‘My hair will dry faster than my hat.’

– Howard Whitley Eves, In Mathematical Circles: Quadrants III and IV, 1969

Math Notes

12 = 3 × 4; 56 = 7 × 8

Words and Numbers

http://commons.wikimedia.org/wiki/File:Number-line.gif

In English, the name of each integer shares a letter with each of its neighbors. ONE shares an O with TWO, TWO shares a T with THREE … and so on to infinity.

Huth’s Moving Star

http://www.sxc.hu/photo/172339

In late 1801, Johann Bode, director of the Berlin Observatory, received a curious series of letters from astronomer Hofrath Huth in Frankfort-on-the-Oder. On Dec. 2 Huth had seen something new in the sky, “a star with faint reddish light, round, and admitting of being magnified.” But it wasn’t a star: On subsequent nights he watched it drift slowly to the southwest, growing gradually fainter, and by Jan. 6 it had disappeared. Huth concluded that he was watching an object recede from Earth.

Unfortunately, Bode was busy with other things, and the weather was too cloudy for him to confirm Huth’s observations. Also, the positional data that Huth had provided were somewhat poor.

Huth wasn’t a nut: Among other things, he co-discovered Comet Encke in 1805. And Nature noted later that he had alerted Bode to the object in time for the director to witness it himself if the skies had been clear. But as it happened, Huth was the only one to witness the curious object, whatever it was. And, whatever it was, it has not returned.

Still Waters

Gauss’ scientific diary was a great boon to mathematical historians, but his notes could be frustratingly cryptic. On July 10, 1796, he made this entry:

ΕΥΡΗΚΑ! num = Δ + Δ + Δ

He had discovered that every positive integer is the sum of at most three triangular numbers.

Among the 146 entries, two remain completely opaque. On Oct. 11, 1796, Gauss had written:

Vicimus GEGAN.

And on April 8, 1799:

gauss diary entry

No one knows what either of these means — if they had mathematical significance, it was lost with Gauss.

So it goes. Dirichlet was famously uncommunicative, not even informing his family that his wife had given birth. His father-in-law later complained that he “should at least have been able to write 2 + 1 = 3.”

Fitting

In the Dewey decimal system, books on number theory are labeled 512.81.

512 = 29 and 81 = 92.

Misc

  • Jimmy Carter was the first U.S. president born in a hospital.
  • Hamlet has 1506 lines, fully 39 percent of the play.
  • 736 = 7 + 36
  • NOOK combines two antonyms.
  • “Everything that deceives may be said to enchant.” — Plato

Symmetric Milestones

What’s unusual about these numbers?

palindromic primes in arithmetic progression 1

Each series is spaced evenly on the number line:

palindromic primes in arithmetic progression 2

Each number is a palindrome.

And each is prime.

Education Reconsidered

Reflect, Socrates; you may have to deny your words.

I have reflected, I said; and I shall never deny my words.

Well, said he, and so you say that you wish Cleinias to become wise?

Undoubtedly.

And he is not wise as yet?

At least his modesty will not allow him to say that he is.

You wish him, he said, to become wise, and not to be ignorant?

That we do.

You wish him to be what he is not, and no longer to be what he is?

I was thrown into consternation at this.

Taking advantage of my consternation he added: You wish him no longer to be what he is, which can only mean that you wish him to perish. Pretty lovers and friends they must be who want their favourite not to be, or to perish!

— Plato, Euthydemus

Math Notes

9 + 9 = 18; 9 × 9 = 81
24 + 3 = 27; 24 × 3 = 72
47 + 2 = 49; 47 × 2 = 94
497 + 2 = 499; 497 × 2 = 994

Two Milestones

The date 11/19/1999 contained only odd digits. Less than three months later, 2/2/2000 contained only even.

That’s a rare coincidence. It had been 1111 years since the last all-even date … and it’ll be 1111 more before the next all-odd one.

Skyward

In 1907, Massachusetts physician Duncan MacDougall conceived a singular experiment. When he observed that a patient at his Haverhill hospital was nearing death, he installed him in a specially constructed bed in his office and measured his weight both before and after death. With six such weighings he determined that humans lose between 0.5 and 1.5 ounces at death.

“Is the soul substance?” he wrote. “It would seem to me to be so. … Here we have experimental demonstration that a substance capable of being weighed does leave the human body at death.”

Similar experiments with 15 dogs showed no change in mass, proving, he decided, that dogs have no souls. MacDougall’s findings were written up briefly in the New York Times and occasioned a flurry of correspondence in American Medicine, but after that they were largely forgotten. But who knows? Perhaps he was right.

Math Notes

math notes

Pop Physics

A can of Diet Coke floats in water, while regular Coke sinks:

Why? The Diet Coke contains 190 mg of aspartame, but the regular Coke contains 39 grams of sugar. So the regular Coke is denser.

E Pluribus Unum

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111 is prime.

Market Forces

The following question was a favourite topic for discussion, and thousands of the acutest logicians, through more than one century, never resolved it: ‘When a hog is carried to market with a rope tied about its neck, which is held at the other end by a man, whether is the hog carried to market by the rope or the man?’

— Isaac Disraeli, Curiosities of Literature, 1893

The Misfortune Field

One of the most enduring contributions to the [Wolfgang] Pauli legend was the ‘Pauli Effect,’ according to which Pauli could, by his mere presence, cause laboratory accidents and catastrophes of all kinds. Peierls informs us that there are well-documented instances of Pauli’s appearance in a laboratory causing machines to break down, vacuum systems to spring leaks, and glass apparatus to shatter. Pauli’s destructive spell became so powerful that he was credited with causing an explosion in a Göttingen laboratory the instant his train stopped at the Göttingen station.

– William H. Cropper, Great Physicists, 2004

(To exaggerate the effect, Pauli’s friends once arranged to have a chandelier crash to the floor when he arrived at a reception. When he appeared, a pulley jammed, and the chandelier refused to budge.)

The Mirror Problem

http://commons.wikimedia.org/wiki/File:Aliceroom3.jpg

In 1868, 8-year-old Alice Raikes was playing with friends in her London garden when a visitor at a neighbor’s house overheard her name and called to her.

“So you are another Alice,” he said. “I’m very fond of Alices. Would you like to come and see something which is rather puzzling?” He led them into a room with a tall mirror in one corner.

‘Now,’ he said, giving me an orange, ‘first tell me which hand you have got that in.’ ‘The right,’ I said. ‘Now,’ he said, ‘go and stand before that glass, and tell me which hand the little girl you see there has got it in.’ After some perplexed contemplation, I said, ‘The left hand.’ ‘Exactly,’ he said, ‘and how do you explain that?’ I couldn’t explain it, but seeing that some solution was expected, I ventured, ‘If I was on the other side of the glass, wouldn’t the orange still be in my right hand?’ I can remember his laugh. ‘Well done, little Alice,’ he said. ‘The best answer I’ve had yet.’

“I heard no more then, but in after years was told that he said that had given him his first idea for Through the Looking-Glass, a copy of which, together with each of his other books, he regularly sent me.”

Bootstraps Everlasting

I recently visited an Eastern sage and asked him, ‘Is it possible to live for ever?’ ‘Certainly,’ he replied, ‘You must undertake to do two things.’ ‘What are they?’ ‘Firstly, you must never again make any false statements.’ ‘That’s simple enough. What is the second thing I must do?’ ‘Every day you must utter the statement “I will repeat this statement tomorrow.” If you follow these instructions faithfully you are certain to live forever.’

— Jacqueline Harman, letter to the Daily Telegraph, Oct. 8, 1985

Misc

  • Q is the only letter that does not appear in any U.S. state name.
  • 6455 = (64 – 5) × 5
  • North Dakota’s record high temperature (121°F) is higher than Florida’s (109°F).
  • UNNOTICEABLY contains the vowels A, E, I, O, and U in reverse order.
  • “An odd thought strikes me: We shall receive no letters in the grave.” — Samuel Johnson

No Vacancy

http://commons.wikimedia.org/wiki/File:Population_curve.svg

The world population has doubled between:

  • 1181 and 1715
  • 1715 and 1881
  • 1881 and 1960
  • 1960 and 1999

It’s expected to reach 9 billion by 2040.

Pandigital Squares

Square numbers containing all 10 digits unrepeated:

320432 = 1026753849
322862 = 1042385796
331442 = 1098524736
351722 = 1237069584
391472 = 1532487609
456242 = 2081549376
554462 = 3074258916
687632 = 4728350169
839192 = 7042398561
990662 = 9814072356

The Mirror

From Albert Beiler, Recreations in the Theory of Numbers (1964):

1 + 4 + 5 + 5 + 6 + 9 = 3 + 2 + 3 + 7 + 8 + 7

Pair each digit on the left with one on the right (for example, 13, 42, 53, 57, 68, 97). The sum of these six numbers will always equal its mirror image:

13 + 42 + 53 + 57 + 68 + 97 = 79 + 86 + 75 + 35 + 24 + 31

This works for all 720 possible combinations.

Most remarkably, you can square every term in these equations and they still hold:

132 + 422 + 532 + 572 + 682 + 972 = 792 + 862 + 752 + 352 + 242 + 312