Science & Math

Misc

  • The telephone number 266-8687 spells both AMOUNTS and CONTOUR.
  • 38856 = (38 – 85) × 6
  • CARTHORSE is an anagram of ORCHESTRA.
  • The French for paper clip is trombone.
  • “The oldest books are only just out to those who have not read them.” — Samuel Butler

Gilbreath’s Conjecture

Doodling on a napkin in 1958, mathematician Norman L. Gilbreath noticed something odd. First he wrote down the first few prime numbers in a row. Then, on each succeeding row, he recorded the (unsigned) difference between each pair of numbers in the row above:

gilbreath's conjecture

The first digit in each row (except the first) is 1. Will this always be true, no matter how many prime numbers we start with? It’s been borne out in computer searches extending to hundreds of billions of rows. But no one knows for sure.

Misc

misc

  • Tarzan’s yell is an aural palindrome.
  • CONTAMINATED is an anagram of NO ADMITTANCE.
  • The Swiss Family Robinson have no surname (“Robinson” refers to Robinson Crusoe).
  • x2 – 2999x + 2248541 produces 80 primes from x = 1460 to 1539.
  • “A great fortune is a great slavery.” — Seneca

Person to Person

Suppose that you enter a cubicle in which, when you press a button, a scanner records the states of all the cells in your brain and body, destroying both while doing so. This information is then transmitted at the speed of light to some other planet, where a replicator produces a perfect organic copy of you. Since the brain of your Replica is exactly like yours, it will seem to remember living your life up to the moment when you pressed the button, its character will be just like yours, and it will be in every other way psychologically continuous with you. Is it you?

– Derek Parfit, “Divided Minds and the Nature of Persons,” in Mindwaves, 1987

Math Notes

multigrade

Cast of Thought

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

Suppose, therefore, a person to have enjoyed his sight for thirty years, and to have become perfectly acquainted with colours of all kinds, except one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be placed before him, descending gradually from the deepest to the lightest; it is plain, that he will perceive a blank, where that shade is wanting, and will be sensible, that there is a greater distance in that place between the contiguous colours than in any other. Now I ask, whether it be possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, though it had never been conveyed to him by his senses?

– David Hume, An Enquiry Concerning Human Understanding, 1748

A Penny Saved

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

When German physicist Walther Nernst learned that his cowshed was warm because of the cows’ metabolic activity, he resolved to sell them and invest in carp.

A thinking man, he said, cultivates animals that are in thermodynamic equilibrium with their surroundings and does not waste his money in heating the universe.

“Plane Geometry”

‘Twas Euclid, and the theorem pi
Did plane and solid in the text,
All parallel were the radii,
And the ang-gulls convex’d.

“Beware the Wentworth-Smith, my son,
And the Loci that vacillate;
Beware the Axiom, and shun
The faithless Postulate.”

He took his Waterman in hand;
Long time the proper proof he sought;
Then rested he by the XYZ
And sat awhile in thought.

And as in inverse thought he sat
A brilliant proof, in lines of flame,
All neat and trim, it came to him,
Tangenting as it came.

“AB, CD,” reflected he–
The Waterman went snicker-snack–
He Q.E.D.-ed, and, proud indeed,
He trapezoided back.

“And hast thou proved the 29th?
Come to my arms, my radius boy!
O good for you! O one point two!”
He rhombused in his joy.

‘Twas Euclid, and the theorem pi
Did plane and solid in the text;
All parallel were the radii,
And the ang-gulls convex’d.

– Emma Rounds

Reservation Trouble

Socrates likes company. He wants to eat only if Plato wants to eat.

But Plato is angry at Socrates. He wants to eat only if Socrates does not want to eat.

Does Socrates want to eat?

(From Buridan’s Sophismata.)

À La Carte

Suppose there were an experience machine that would give you any experience you desired. Superduper neuropsychologists could stimulate your brain so that you would think and feel you were writing a great novel, or making a friend, or reading an interesting book. All the time you would be floating in a tank, with electrodes attached to your brain. Should you plug into this machine for life, preprogramming your life’s experiences? If you are worried about missing out on desirable experiences, we can suppose that business enterprises have researched thoroughly the lives of many others. You can pick and choose from their large library or smorgasbord of such experiences, selecting your life’s experiences for, say, the next two years. After two years have passed, you will have ten minutes or ten hours out of the tank, to select the experiences of your next two years. Of course, while in the tank you won’t know that you’re there; you’ll think it’s all actually happening. … Would you plug in?

– Robert Nozick, Anarchy, State, and Utopia, 1974

Trivium

10102323454577 is the smallest 14-digit prime number that follows the rhyme scheme of a Shakespearean sonnet (ababcdcdefefgg).

(Discovered by Jud McCranie.)

Misc

  • Richard Gere’s middle name is Tiffany.
  • Where does the hinterland begin?
  • WORLD CUP TEAM is an anagram of TALCUM POWDER.
  • log 237.5812087593 = 2.375812087593
  • “Why is it that something can be transparent green but not transparent white?” — Wittgenstein

Round Trip

chandler perpetual motion

Eric Chandler offered this perpetual-motion scheme for Edward Barbeau’s “Fallacies, Flaws and Flimflam” column in the College Mathematical Journal. Points A and B are at the same height, and C is halfway between them. The ramp AC is a segment of a cycloid, a curve designed to produce the fastest descent under gravity.

A ball released at A rolls down the ramp AC to C covering a greater distance in a shorter time than it would have had it rolled down BC to C. The relation Velocity = Distance/Time thus implies that the ball arrives at C with greater velocity than it would have had it rolled down BC. This added velocity enables the ball to roll from C up to and past B to a point D a little farther along. It then returns to A along the inclined ramp DA to repeat the cycle endlessly.

Where is the error?

Math Notes

99999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999989999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999999999999999999999 is prime.

Unforgettable

In probability theory, the formula for the Poisson distribution is

Pm(n) = mne-m/n!

Pleasingly, the mnemonic for this is mnemonic: “m to the n, e to the -m over n factorial.” Arguably the factorial sign even resembles an inverted i.

Now we just need a way to remember that

(From M.H. Greenblatt, Mathematical Entertainments, 1965.)

Case Closed

There was Diodorus Chronos, a most acute and subtle reasoner. He proved there was no such thing as motion. A body must move either in the place where it is or in the place where it is not. Now, a body cannot be in motion in the place where it is stationary, and cannot be in motion in the place where it is not. Therefore, it cannot move at all. …

Diodorus was brought up roundly by another densely practical intelligence. Having dislocated his shoulder, he sent for a surgeon to set it. ‘Nay,’ said the practitioner, doubtful, perhaps, whether so subtle an intelligence might not euchre him out of his fee by some logical ingenuity, ‘your shoulder cannot possibly be put out at all, since it cannot be put out in the place in which it is, nor yet in the place in which it is not.’

– “Some Famous Paradoxes,” The Illustrated American, Nov. 1, 1890

The Flying Bird

loyd flying bird problem

A correspondent at Princeton College sent this conundrum to Sam Loyd:

“Supposing that a bird weighing one ounce flies into a box with only one small opening, and without resting continues to fly round and round in the box, would it increase or lessen the weight of the box?”

Loyd said he was open to argument, but “the preponderance of opinion is so overwhelmingly in favor of the weight of the bird being added to that of the box, that it would be difficult to present reasonable argument for the other side, despite of the popular belief that such would be the case. … The bird is heavier than the air and supports itself by striking down upon the air and the power of such strokes would undoubtedly show on the dial the difference in weight between the bird and its displacement of air.”

A related problem from Clark Kinnaird’s Encyclopedia of Puzzles and Pastimes (1946):

“A vagrant who stole three melons weighing three pounds each, came to a bridge which was just strong enough to hold him and six pounds. Without throwing any of the melons across the bridge, how did the vagrant cross the bridge with the melons, none of which touched the bridge?”

Kinnaird’s answer: He juggled them.

Math Notes

1212 + 1388 + 2349 = 4949; 49493 = 121213882349
1287 + 1113 + 2649 = 5049; 50493 = 128711132649
1623 + 2457 + 1375 = 5455; 54553 = 162324571375
1713 + 2377 + 1464 = 5554; 55543 = 171323771464
3689 + 1035 + 2448 = 7172; 71723 = 368910352448

Applied Chemistry

When Hitler’s army marched into Copenhagen, Niels Bohr had to decide how to safeguard the Nobel medals of James Franck and Max von Laue, which they had entrusted to him. Sending gold out of the country was almost a capital offense, and the physicists’ names were engraved on the medals, making such an attempt doubly risky. Burying the medals seemed uncertain as well. Finally his friend the Hungarian physicist Georg von Hevesy invented a novel solution: He dissolved the medals in a jar of aqua regia, which Bohr left on a shelf in his laboratory while he fled to Sweden.

When he returned in 1945, the jar was still there. Bohr had the gold recovered, and the Nobel Foundation recast it into two medals.

(Chemist Hermann Mark found a way to escape Germany with his money: He used it to buy platinum wire, which he fashioned into coat hangers. Once he had brought these successfully through customs, he sold them to recover the money.)

Twice Told

Do those, said he, who learn, learn what they know, or what they do not know?

Cleinias had answered Euthydemus that those who learned learn what they do not know; and he put him through a series of questions the same as before.

Do you not know letters?

He assented.

All letters?

Yes.

But when the teacher dictates to you, does he not dictate letters?

To this also he assented.

Then if you know all letters, he dictates that which you know?

This again was admitted by him.

Then, said the other, you do not learn that which he dictates; but he only who does not know letters learns?

Nay, said Cleinias; but I do learn.

Then, said he, you learn what you know, if you know all the letters?

He admitted that.

Then, he said, you were wrong in your answer.

– Plato, Euthydemus

The Two Cultures

Paul Dirac, the British theoretical physicist, has a reputation for being reserved and speaking little. He read E.M. Forster’s A Passage to India, and commented favorably on it. Someone in Cambridge thought the two great men ought to meet. J.G. Crowther recalls that this was arranged. The two men observed each other in long, silent respect. Presently Dirac asked, ‘What happened in the cave?’ ‘I don’t know,’ said Forster, which concluded their conversation.

Bulletin of the Atomic Scientists, March 1971

(Elsewhere Dirac gave his opinion of Crime and Punishment — “He describes a sunset, and then a little later the same evening the sun sets again. That kind of mistake does jar on me.”)

The Dr. Psycho Paradox

You’re eating apples with your friend Dr. Psychic Psycho, a talented biochemist who fancies himself a clairvoyant and has made many accurate oddball predictions.

“I have interesting news for you,” he says. “You must seriously consider taking this pill. As you know (since we have recently determined it together), it contains substance X, which (as you also know — but consult this pharmacopeia if in doubt) is fatally poisonous by itself, while nevertheless furnishing unfailing antidote to poison Z — though it does have some minorly unpleasant side effects. Now the apple I gave to you, which you have just finished eating, was poisoned by me with Z — or not — in line with my prediction as to your taking or not taking the antidote pill. Benign old me, of course, only poisoned the apple if I foresaw that you were indeed going to take the antidote. And not to worry — I’m a very good predictor.”

He rushes off. What should you do?

(By University of Pittsburgh philosopher Nicholas Rescher.)

Stare Conditioning

In 1958, B.F. Skinner and Erich Fromm attended the same California symposium. Skinner found that Fromm “proved to have something to say about almost everything, but with little enlightenment,” and “when he began to argue that people were not pigeons, I decided that something had to be done”:

On a scrap of paper I wrote ‘Watch Fromm’s left hand. I am going to shape a chopping motion’ and passed it down the table to [Halleck Hoffman]. Fromm was sitting directly across from the table and speaking mainly to me. I turned my chair slightly so that I could see him out of the corner of my eye. He gesticulated a great deal as he talked, and whenever his left hand came up, I looked straight at him. If he brought the hand down, I nodded and smiled. Within five minutes he was chopping the air so vigorously that his wristwatch kept slipping out over his hand.

“William Lederer had seen my note, and he whispered to Halleck. The note came back with an addendum: ‘Let’s see you extinguish it.’ I stopped looking directly across the table, but the chopping went on for a long time. It was an unfair trick, but Fromm had angered me — first with his unsupported generalizations about human behavior and then with the implication that nothing better could be done if ‘people were regarded as pigeons.’”

(From Skinner’s 1983 memoir A Matter of Consequences.)

Math Notes

kordemsky multigrade