Science & Math

Conjuring

Remove any nine cards from an ordinary deck, shuffle them, and deal them face down into three piles. Choose any pile and note its bottom card. Then assemble the three piles into one, being sure to place the chosen pile on top.

Suppose the card you chose is the three of spades. Spell T-H-R-E-E, dealing one card face down onto the table with each letter. Place the remaining cards on top of these five and take up the whole packet. Now spell O-F, and again place the remaining cards on top of these two. Then spell S-P-A-D-E-S and place the remaining cards on top.

Now pick up the packet and spell M-A-G-I-C, dealing the final card face up. It’s the three of spades.

Remarkably, this trick will produce any card, from the 10-letter ace of clubs to the 15-letter queen of diamonds. It was invented by California magician Jim Steinmeyer and appears in his 2002 book Impuzzibilities (used by permission).

Misc

  • Mississippi didn’t ratify the 13th Amendment, abolishing slavery, until 2013.
  • To protect its ecosystem, the location of Hyperion, the world’s tallest living tree, is kept secret.
  • 34425 = 34 × 425
  • CIRCUMSTANTIAL EVIDENCE = ACTUAL CRIME ISN’T EVINCED
  • “Well, if I called the wrong number, why did you answer the phone?” — James Thurber

Surface Tension

Lewis Carroll on the perils of physics:

Suppose a solid held above the surface of a liquid and partially immersed: a portion of the liquid is displaced, and the level of the liquid rises. But, by this rise of level, a little bit more of the solid is of course immersed, and so there is a new displacement of a second portion of the liquid, and a consequent rise of level. Again, this second rise of level causes a yet further immersion, and by consequence another displacement of liquid and another rise. It is self-evident that this process must continue till the entire solid is immersed, and that the liquid will then begin to immerse whatever holds the solid, which, being connected with it, must for the time be considered a part of it. If you hold a stick, six feet long, with its end in a tumbler of water, and wait long enough, you must eventually be immersed. The question as to the source from which the water is supplied — which belongs to a high branch of mathematics, and is therefore beyond our present scope — does not apply to the sea. Let us therefore take the familiar instance of a man standing at the edge of the sea, at ebb-tide, with a solid in his hand, which he partially immerses: he remains steadfast and unmoved, and we all know that he must be drowned.

“The multitudes who daily perish in this manner to attest a philosophical truth, and whose bodies the unreasoning wave casts sullenly upon our thankless shores, have a truer claim to be called the martyrs of science than a Galileo or a Kepler.”

Round Trip

http://www.google.com/patents/US5678489

Frustrated in trying to describe higher topology abstractly to students, Xian Wang invented a model train that can hug either side of a track:

It is therefore a primary object of the present invention to provide an electrically-operated ornament travelling on a rail which can be used to explain the Mobius Theorem. … In general textbooks, this advanced mathematic rule is usually explained by demonstrating a body circularly moving on a front and a reverse side of a twisted two-ends-connected belt. Most people can not understand and imagine the theorem from such explanation and demonstration.

Of course, once you’ve built one you can put it to other uses:

http://www.google.com/patents/US5678489

Morphology

nabokov on kafka

This is the opening page of “The Metamorphosis,” from Vladimir Nabokov’s teaching copy. Kafka’s novella held a special interest for Nabokov, who was a trained entomologist. From his lecture notes at Cornell:

Now, what exactly is the ‘vermin’ into which poor Gregor, the seedy commercial traveler, is so suddenly transformed? It obviously belongs to the branch of ‘jointed leggers’ (Arthropoda), to which insects, and spiders, and centipedes, and crustaceans belong. … Next question: What insect? Commentators say cockroach, which of course does not make sense. A cockroach is an insect that is flat in shape with large legs, and Gregor is anything but flat: he is convex on both sides, belly and back, and his legs are small. He approaches a cockroach in only one respect: his coloration is brown. That is all. Apart from this he has a tremendous convex belly divided into segments and a hard rounded back suggestive of wing cases. In beetles these cases conceal flimsy little wings that can be expanded and then may carry the beetle for miles and miles in a blundering flight. … Further, he has strong mandibles. He uses these organs to turn the key in a lock while standing erect on his hind legs, on his third pair of legs (a strong little pair), and this gives us the length of his body, which is about three feet long. … In the original German text the old charwoman calls him Mistkafer, a ‘dung beetle.’ It is obvious that the good woman is adding the epithet only to be friendly. He is not, technically, a dung beetle. He is merely a big beetle.

“Curiously enough,” he added, “Gregor the beetle never found out that he had wings under the hard covering of his back. This is a very nice observation on my part to be treasured all your lives. Some Gregors, some Joes and Janes, do not know that they have wings.”

Square Deal

square deal - problem

A “coffin,” or killer problem, from the oral entrance exams to the math department of Moscow State University:

Construct (with ruler and compass) a square given one point from each side.

Click for Answer

The Shooting Room

You’re about to play a game. A single person enters a room and two dice are rolled. If the result is double sixes, he is shot. Otherwise he leaves the room and nine new players enter. Again the dice are rolled, and if the result is double sixes, all nine are shot. If not, they leave and 90 new players enter.

And so on, the number of players increasing tenfold with each round. The game continues until double sixes are rolled and a group is executed, which is certain to happen eventually. The room is infinitely large, and there’s an infinite supply of players.

If you’re selected to enter the room, how worried should you be? Not particularly: Your chance of dying is only 1 in 36.

Later your mother learns that you entered the room. How worried should she be? Extremely: About 90 percent of the people who played this game were shot.

What does your mother know that you don’t? Or vice versa?

(Paul Bartha and Christopher Hitchcock, “The Shooting Room Paradox and Conditionalizing on Measurably Challenged Sets,” Synthese, March 1999)

(Thanks, Zach.)

Pièce de Résistance

Take an ordinary magic square and replace its numbers with resistors of the same ohmic value. Now the set of resistors in each row, column, and diagonal will yield the same total resistance value when joined together end to end.

This paramagic square, by Lee Sallows, is similar — except that the resistors must be joined in parallel:

sallows paramagic square

The Problem of the Two Boys

A family has two children, and you know that at least one of them is a boy. What is the probability that both are boys? There are four possibilities altogether (boy-boy, boy-girl, girl-boy, and girl-girl), and we can eliminate the last, so it would seem that the answer is 1/3.

But now suppose you visit a family that you know has two children, and that a boy comes into the room. What is the probability that both children are boys? Of the two children, you know that this one is a boy, and there is a probability of 1/2 that the other is a boy. So it seems that there is a probability of 1/2 that both are boys.

How can this be? We seem to have the same amount of information in both cases. Why does it lead us to two different conclusions?

Din Minimum

In 1958, acoustician William MacLean of the Polytechnic Institute of Brooklyn answered a perennial question: How many guests can attend a cocktail party before it becomes too noisy for conversation? He declared that the answer, for a given room, is

cocktail party noise

where

N0 = the critical number of guests above which each speaker will try overcome the background noise by raising his voice
K = the average number of guests in each conversational group
a = the average sound absorption coefficient of the room
V = the room’s volume
h = a properly weighted mean free path of a ray of sound
d0 = the conventional minimum distance between speakers
Sm = the minimum signal-to-noise ratio for the listeners

When the critical guest N0 arrives, each speaker is forced to increase his acoustic power in small increments (“I really don’t know what she sees in him.” — “Beg your pardon?” — “I say, I REALLY DON’T KNOW WHY SHE GOES OUT WITH HIM”) until each group is forced to huddle uncomfortably close in order to continue the conversation.

“We see therefore that, once the critical number of guests is exceeded, the party suddenly becomes a loud one,” MacLean concluded, somewhat sadly. “The power of each talker rises exponentially to a practical maximum, after which each reduces his or her talking distance below the conventional distance and then maintains, servo fashion, just the proximity, tête à tête, required to attain a workable signal-to-noise ratio. Thanks to this phenomenon the party, although a loud one, can still be confined within one apartment.”

(William R. MacLean, “On the Acoustics of Cocktail Parties,” Journal of the Acoustical Society of America, January 1959, 79-80.)

Construction Set

From the ever-inventive Lee Sallows, a self-tiling tile set:

sallows self-tiling tiles

His article on such self-similar tilings appears in the December 2012 issue of Mathematics Magazine.

Vanishing Act

In 1913 mathematician P.E.B. Jourdain proposed a familiar paradox:

On one side of a blank strip of paper, write The statement on the other side of this paper is true.

On the other side, write The statement on the other side of this paper is false.

“The paradox in this form is quite vulnerable to an absolute refutation,” wrote Valdis Augstkalns in a 1970 letter to The Listener. “One takes the paper, gives it a half twist, and joins the ends to form a Möbius strip. The serious and philosophically legitimate question is transformed to ‘Eminent members of the panel, which is the other side of the paper?'”

Topsy Turvy

http://books.google.com/books?id=iUoXAAAAYAAJ

This reversible magic square comes from Henry Dudeney’s Canterbury Puzzles.

Each row, column, and diagonal in the square totals 179.

Thanks to some clever calligraphy, this remains true when the square is turned upside down.

Misc

  • A pound of dimes has the same value as a pound of quarters.
  • The French word hétérogénéité has five accents.
  • 32768 = (3 – 2 + 7)6 / 8
  • Can you deceive yourself deliberately?
  • “My country is the world, and my religion is to do good.” — Thomas Paine

In 2000, Guatemalan police asked Christmas revelers not to fire pistols into the air. “Lots of people die when bullets fall on their heads,” National Civilian Police spokesman Faustino Sanchez told Reuters. He said that five to ten Guatemalans are killed or injured each Christmas by falling bullets.

The Chosen Ones

http://commons.wikimedia.org/wiki/File:Hans_Hoffmann_nach_D%C3%BCrer_Hirschk%C3%A4fer.jpg

J.B.S. Haldane was once asked what his study of biology had taught him about God.

He said that the Creator, if he exists, has “an inordinate fondness for beetles.”

Number Squares

Howard W. Bergerson devised these:

number squares

As in a word square, the “columns” in each sum are identical with the “rows.”

Gene Pool

Asked whether he would give his life to save a drowning brother, J.B.S. Haldane said, “No, but I would to save two brothers or eight cousins.”

(Whack)

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

In 1877, five-year-old Bertrand Russell asked his Aunt Agatha, “Aunty, do limpets think?”

She said, “I don’t know.”

He said, “Then you must learn.”

Misc

  • Will Rogers died at the northernmost point in the United States.
  • 94122 + 23532 = 94122353
  • TO BE OR NOT TO BE contains two Bs.
  • If you stop me being mute, what sound do I make?
  • “Better to ask twice than to lose your way once.” — Danish proverb

Inside Out

http://www.google.com/patents/US6419111

Erl E. Kepner patented a bewildering object in 2002 — a one-sided coffee mug:

To help us see the unique properties of the beverage vessel, let’s pretend that the vessel is made of astonishingly thin material. This is a ‘thought experiment’, not a real experiment, where you actually physically do anything. Note that the only edge on the vessel is the rim that your lips would touch if you drank coffee from it. The rim of the container would be a very sharp edge while the areas where the container and the hollow handle come together would be smooth curved shapes. Now pretend that you have a very tiny little black ball shaped magnet located on the surface of the vessel somewhere and another little tiny white ball magnet located on the opposite side of the vessel material. If one were to (mentally) move either the black or white ball magnet, it would cause the white or black ball magnet to move also. One could move the black or white ball magnets, one at a time, along the vessel surface so that white ball magnet ended up where the black ball magnet was initially located and the black ball magnet was where the white ball magnet originally was. This can be accomplished without having either of the magnets pass over the vessel rim, which is the only edge of the vessel. Other than the Klein Bottle, no other hollow shape has this property. Another way to envision or demonstrate the unique properties of this shape is to point out the fact that a little bug can crawl from any point on the surface of the vessel to any other point on the surface of the vessel without crossing over an edge. Bugs cannot do this on a normal coffee cup or any other three-dimensional shape that we use in our daily lives.

“The future marketing of the beverage container of this invention will use these sorts of interesting points to stimulate interest among technically well educated people and everyday people with an innate curiosity and appreciation for the wonder and beauty of mathematics and nature.”

See Carry-All and the Klein bottle recycling center.

The Perfect Crime

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

Suspect A has shot a man through the heart during the last half minute. But Suspect B shot him through the heart during the preceding 1/4 minute, Suspect C shot him through the heart during the 1/8 minute before that, and so on. Assuming that a bullet through the heart kills a man instantly, the victim must already have been dead before any given suspect shot him.

Indeed, notes José Benardete, he cannot be said to have died of a bullet wound.

“The Research Man’s Prayer”

Help me be MANIC so I may be joyous though the results are equivocal.

Help me be DEPRESSIVE for when a prediction is verified, I must know that it will not later be confirmed.

Help me be SADISTIC so I suffer not though the subjects be sorely anguished.

Help me be MASOCHISTIC for even the most obstinate experimental animal should be a pleasure to me.

Help me be PSYCHOPATHIC to quiet the guilt when I tell loved ones that the experiment is going well.

Help me be SCHIZOPHRENIC to sustain myself by finding hopeful trends in random data.

Help me be PARANOID so I can see in the hostile attitudes of others the supremacy of my own work.

Help me to have ANXIETY ATTACKS so that even on holidays I find myself toiling in the laboratory.

And finally,

Help my wife get a job! for when I cross over the shadowy border of normalcy, somebody will have to support the kids. Amen.

— R.A McCleary in the Worm Runner’s Digest, November 1960

Double Trouble

The properties of the simple Möbius strip are well understood: Take a strip of paper, give it a half-twist, and tape the ends together. Now an ant can traverse the full length of the loop, on both sides, and return to its starting point without ever crossing an edge.

But try doing the same thing with two strips of paper. Pair the strips, give them a half-twist, and connect the ends. Now it’s possible to insert a toothpick between the bands and to draw the toothpick along the entire length of the loop, which seems to show that they’re two distinct objects. But if you draw a line along either strip, starting anywhere, you’ll find that you traverse both strips and return to your starting point.

“I have known people to ponder this for hours while listening to Pink Floyd without ever fully appreciating what they have beheld,” writes Clifford Pickover in The Möbius Strip. Are you holding one object or two?

The Paradox of the Second Ace

http://commons.wikimedia.org/wiki/File:Bertall_-_Le_whist_3.jpg

You’re watching four statisticians play bridge. After a hand is dealt, you choose a player and ask, “Do you have at least one ace?” If she answers yes, the chance that she’s holding more than one ace is 5359/14498, which is less than 37 percent.

On a later hand, you choose a player and ask, “Do you have the ace of spades?” Strangely, if she says yes now the chance that she has more than one ace is 11686/20825, which is more than 56 percent.

Why does specifying the suit of her ace improve the odds that she’s holding more than one ace? Because, though a smaller number of potential hands contain that particular ace, a greater proportion of those hands contain a second ace. It’s counterintuitive, but it’s true.