Science & Math

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

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111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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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

Perpetual Notion

http://commons.wikimedia.org/wiki/File:Perpetuum1.png

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.

Hendecadivisibility

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