Science & Math

Say It With Constants

MIT mails its acceptance letters on 3/14 — “Pi Day.”

The Life on Mars Paradox

http://commons.wikimedia.org/wiki/Image:Mars_atmosphere.jpg

What’s the probability that there are horses on Mars? Let’s be extremely generous and say it’s 1/2. And let’s also say there’s a probability of 1/2 that there are parrots, and aardwolves, and each of 17 other species.

Then the probability that none of these 20 species exists is (1/2)20. And the probability that at least one of them exists is 1 – (1/2)20, or 0.999999046.

Thus it’s nearly certain that there’s life on Mars.

Witchcraft

Mersenne once wrote to Fermat asking whether 100895598169 were a prime number.

Fermat replied immediately that it’s the product of 898423 and 112303, both of which are prime.

To this day, no one knows how he knew this. Has a powerful factoring technique been lost?

Math Notes

93 + 13 + 93 = 1459
13 + 43 + 53 + 93 = 919

How to Win Six Million Dollars

http://www.sxc.hu/photo/276465

Summon six millionaires and invite them to stake their fortunes on a single hand of poker. They will eagerly agree. Open a new deck of cards, discard the jokers, and ask the millionaires to cut (but not shuffle!) the deck as many times as they like. Then deal seven hands, ostentatiously dealing your own second and fourth cards from the bottom of the deck.

The millionaires may be reluctant to object to this, as all six of them will be holding full houses. (This works — try it.) But “See here,” they will finally say. “What was that business with the bottom-dealing? You’re up to something. We insist that you discard that hand.” Look hurt, then deal yourself a new hand.

You’ll likely be holding a straight flush.

Showoff

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

English mathematician John Wallis (1616-1703) had an odd way of passing time:

“I note that on 22 December, 1669, he, when in bed, occupied himself in finding (mentally) the integral part of the square root of 3 × 1040; and several hours afterwards wrote down the result from memory. This fact having attracted notice, two months later he was challenged to extract the square root of a number of fifty-three digits; this he performed mentally, and a month later he dictated the answer, which he had not meantime committed to writing.”

— W.W. Rouse Ball, Mathematical Recreations & Essays, 1892

Occupational Hazards

Excerpts from the log of J.E. Duane, a weather observer at Long Key, Fla., when the most intense hurricane in U.S. history made landfall on Sept. 2, 1935:

9:20 p.m. I put my flashlight out to sea and could see walls of water which seemed many feet high. I had to race fast to regain the entrance of the cottage, but water caught me waist deep, although writer was only about 60 feet from doorway of cottage. Water lifted cottage from foundations and it floated.

10:15 p.m. The first blast from SSW, full force. House breaking up — wind seemed stronger than any time during storm. I glanced at barometer which read 26.98 inches, dropped it in the water and was blown outside into sea; got hung up in broken fronds of coconut tree and hung on for dear life. I was struck by some object and knocked unconscious.

Later: “2:25 a.m. I became conscious in tree and found I was lodged about 20 feet above ground.”

A Physics Problem

You’re in a rowboat in a swimming pool, and you’re holding a cannonball. If you throw the ball into the pool, will the water level rise or fall?

Click for Answer

King Size

In 1989, paleontologists discovered the fragmentary remains of an enormous dinosaur in southern India.

If estimates are accurate, Bruhathkayosaurus was 145 feet long and weighed 240 tons.

The largest modern whale is 110 feet long and weighs 195 tons.

Math Notes

32 + 42 = 52

212 + 222 + 232 + 242 = 252 + 262 + 272

Bike Trip

The first LSD trip took place on April 19, 1943, when Swiss chemist Albert Hofmann ingested 250 micrograms and tried to go home:

I had to struggle to speak intelligibly. I asked my laboratory assistant, who was informed of the self-experiment, to escort me home. We went by bicycle, no automobile being available because of wartime restrictions on their use. On the way home, my condition began to assume threatening forms. Everything in my field of vision wavered and was distorted as if seen in a curved mirror. I also had the sensation of being unable to move from the spot. Nevertheless, my assistant later told me we had traveled very rapidly.

It’s remembered as “Bicycle Day.”

Math Notes

135 = 1 + 32 + 53
175 = 1 + 72 + 53
518 = 5 + 12 + 83
598 = 5 + 92 + 83

Guess

Once upon a time, there lived a rich farmer who had 30 children, 15 by his first wife who was dead, and 15 by his second wife. The latter woman was eager that her eldest son should inherit the property. Accordingly one day she said to him, “Dear Husband, you are getting old. We ought to settle who shall be your heir. Let us arrange our 30 children in circle, and counting from one of them, remove every tenth child until there remains but one, who shall succeed to your estate.”

The proposal seemed reasonable. As the process of selection went on, the farmer grew more and more astonished as he noticed that the first 14 to disappear were children by his first wife, and he observed that the next to go would be the last remaining member of that family. So he suggested that they should see what would happen if they began to count backwards from this lad. She, forced to make an immediate decision, and reflecting that the odds were now 15 to 1 in favour of her family, readily assented. Who became the heir?

16 children

— W.W. Rouse Ball, Mathematical Recreations & Essays, 1892

Shades of Gray

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

An optical illusion. Squares A and B are the same color.

Richard’s Paradox

Clearly there are integers so huge they can’t be described in fewer than 22 syllables. Put them all in a big pile and consider the smallest one. It’s “the smallest integer that can’t be described in fewer than 22 syllables.”

That phrase has 21 syllables.

Math Notes

84 + 24 + 04 + 84 = 8208

Buffon’s Needle

Remarkably, you can estimate π by dropping needles onto a flat surface. If the surface is ruled with lines that are separated by the length of a needle, then:

buffon's needle

drops is the number of needles dropped. hits is the number of needles that touch a line. The method combines probability with trigonometry; a needle’s chance of touching a line is related to the angle at which it comes to rest. It was discovered by the French naturalist Georges-Louis Leclerc in 1777.

Clarke’s Law

Clarke’s Third Law: Any sufficiently advanced technology is indistinguishable from magic.

Benford’s Corollary: Any technology distinguishable from magic is insufficiently advanced.

Raymond’s Second Law: Any sufficiently advanced system of magic would be indistinguishable from a technology.

Sterling’s Corollary: Any sufficiently advanced garbage is indistinguishable from magic.

Langford’s application to science fiction: Any sufficiently advanced technology is indistinguishable from a completely ad-hoc plot device.

Math Notes

13 + 33 + 63 = 244
23 + 43 + 43 = 136

The Necktie Paradox

http://www.sxc.hu/photo/599259

You and I are having an argument. Our wives have given us new neckties, and we’re arguing over which is more expensive.

Finally we agree to a wager. We’ll ask our wives for the prices, and whoever is wearing the more expensive tie has to give it to the other.

You think, “The odds are in my favor. If I lose the wager, I lose only the value of my tie. If I win the wager, I gain more than the value of my tie. On balance I come out ahead.”

The trouble is, I’m thinking the same thing. Are we both right?

Unquote

http://commons.wikimedia.org/wiki/Image:Julia_set_%28highres_01%29.jpg

“Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is.” — Paul Erdös

Math Notes

73939133
7393913
739391
73939
7393
739
73
7

… are all prime. So are:

357686312646216567629137
57686312646216567629137
7686312646216567629137
686312646216567629137
86312646216567629137
6312646216567629137
312646216567629137
12646216567629137
2646216567629137
646216567629137
46216567629137
6216567629137
216567629137
16567629137
6567629137
567629137
67629137
7629137
629137
29137
9137
137
37
7

But see Not So Fast.

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Viagra keeps plants from wilting.

Israeli and Australian researchers found that a low concentration in water doubled the shelf life of cut flowers, from one week to two weeks.

Recursive Gratitude

Mathematician J.E. Littlewood once wrote a paper for the French journal Comptes Rendus. A Prof. M. Riesz did the translation, and at the end Littlewood found three footnotes:

I am greatly indebted to Prof. Riesz for translating the present paper.

I am indebted to Prof. Riesz for translating the preceding footnote.

I am indebted to Prof. Riesz for translating the preceding footnote.

Littlewood notes that this could have gone on indefinitely but “I stop legitimately at number 3: however little French I know I am capable of copying a French sentence.”