by Garth Sundem
CONSTELLATIONS OF THE NORTHERN HEMISPHERE IN SUMMER
FIVE CLASSIC THOUGHT EXPERIMENTS YOU CAN DO WITHOUT GETTING OUT OF BED
Throughout history, scientists, philosophers, and Ph.D. students lacking funding for actual research have turned to the thought experiment in hopes of discovering something publishable, thereby retaining tenure and/or attracting the admiration of comely undergraduates. The best thought experiments throw light into dark corners of the universe and also provide other scientists, philosophers, and destitute Ph.D. students a way to kill time while waiting for the bus.
MAXWELL'S DEMON
The second law of thermodynamics states that a system will never spontaneously move toward a higher degree of order. It takes energy to increase order.
But imagine you had a box filled with molecules, vibrating away at various speeds and creating by their interaction a constant temperature inside the box. Now stick a divider down the middle of the box, splitting it into two chambers. In this divider is a tiny door operated by a demon. The demon opens and closes the door, allowing faster (hotter) particles to bounce naturally into the right chamber and slower (cooler) particles to bounce into the left chamber. Over time, the order of the system is increased—the right chamber gets hotter and the left chamber gets cooler. Remember, the demon has added or subtracted nothing from the system, only opened and closed a door, thus allowing particles to pass through on their natural paths.
Does this violate the second law of thermodynamics?
SHIP OF THESEUS
The Greek historian Plutarch described the following dilemma: “The ship wherein Theseus and the youth of Athens returned was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their place.” By the time of Demetrius Phalereus, which was about a thousand years after The-seus's return from Crete, so many planks and timbers had been replaced that none of his ship's original wood remained.
The question is, was it still Theseus's ship? More generally, what creates identity? If all the molecules of a thing (or person) are identical to the molecules of another thing (or person) are the two the same? If they are different, what makes them so? If a person were teleported by a machine that disintegrated their molecules and then reassembled them in an exact copy, would it be the same person?
SCHRÖDINGER'S CAT
In 1949, the physicists Erwin Schrödinger and Albert Einstein got together to chat about reality. This led to a number of discoveries, among them the first law of physicist-assisted entropy, which states that whenever two physicists get together to chat about reality, the total amount of reality (R) in the universe is decreased in direct proportion to the combined IQ of said physicists. The FLOPAE was a product mostly of Schrödinger's Cat, a thought experiment in which the feline in question was eventually pronounced both alive and dead at the same time (according to quantum physics …).
First, imagine a cat in a box. You can't see in and the cat can't see out. What the cat can see is a rock, a Geiger counter, and a vial of poison. Now, the rock is slightly radioactive, with an exactly fifty-fifty chance of emitting a subatomic particle in the course of an hour. If the rock emits a particle, the Geiger counter will flip a switch that breaks the vial of poison, killing the cat. (Note to PETA: thought experiment.)
At the end of the hour, from the point of view of an observer outside the box, is the cat alive or dead? Maybe both at the same time? Has this event generated one world in which the cat is alive and another world in which the cat is dead?
Physicist Stephen Hawking said, “When I hear of Schrö-dinger's Cat, I reach for my gun.”
THE CHINESE ROOM
Most proponents of strong artificial intelligence consider John Searle a naysayer. Searle, contrary to every AI-geek's dream, asserts that no matter how much code we write, a computer will never gain sentient understanding. He illustrates his claim with the following example:
Suppose a computer could be programmed to speak Chinese well enough that a Chinese speaker could ask the computer a question and it could respond correctly and idiomatically. The Chinese speaker would not know whether he or she was conversing with a human or a computer, and thus the computer can be said to have humanlike understanding, right?
Wrong, according to Searle. He imagined himself sitting inside the computer, performing the very computerlike function of accepting the input of Chinese characters and then using a system of rules (millions of cross-indexed file cabinets, in his example) to decode these symbols and choose an appropriate response. He could do this for years and years, eventually becoming proficient enough to offer responses correct and idiomatic enough to converse with a native Chinese speaker. Still, he would never actually learn how to speak Chinese.
There are many counterarguments, including the idea that, while Searle himself, sitting inside the computer, doesn't understand Chinese, the system as a whole—Searle, the input system, the filing cabinets, and the output system—does.
What do you think?
ACHILLES AND THE TORTOISE: ZENO'S PARADOX
Here's a classic, pulled straight from the humanities course you took senior year in high school:
Achilles and a tortoise have a race. Achilles, being much the faster, allows the tortoise a hundred-yard head start. Of course, because Achilles allowed the turtle to begin ahead of him, it takes time for Achilles to reach the tortoise's starting point. However, the turtle is no longer there; it has continued, and Achilles must again catch up. Every time Achilles reaches a point the turtle has passed, the turtle has used the time to travel farther ahead. Always having to make up some distance, however small, Achilles will never catch the tortoise!
Right?
SIX CHESS OPENINGS WITH COOL NAMES
Chess is as much art as game. Played at the highest level, it is as much intuition as information. Played at a much lower level, it is about looking cool at coffee shops. Integral to this goal is being able raise an eyebrow at your opponent (who hopefully is wearing a turtleneck while smoking clove cigarettes), and challenge him or her in a voice audible at neighboring tables to counter your Nimzo-Indian Defense. (If you're really cool, you'll explain that the proper spelling is “defence.”) In the boards below, white plays the named defense.
chess champion (and proponent of Russian democracy) Garry Kasparov used the Grünfeld defense in his World Championship matches against Anatoly Karpov in 1986, 1987, and 1990. Kasparov won all three matches. If you want to be as cool as Garry Kasparov, you will use the Grünfeld defense too.
ASIMOV'S THREE LAWS OF ROBOTICS
A robot may not injure a human being or, through inaction, allow a human being to come to harm.
A robot must obey orders given to it by human beings except where such orders would conflict with the First Law.
A robot must protect its own existence as long as such protection does not conflict with the First or Second Law.
THE THREE BASIC PRINCIPLES OF ECONOMICS
SUPPLY AND DEMAND
The phrase “supply and demand” was first used by James Denham-Steuart in his 1767 work Inquiry into the Principles of Political Economy. Little has changed since then—supply and demand still explains that only if people want something will it be supplied and that it will be supplied in quantity commensurate with how much people want. Inefficiency results when supply does not meet demand. If supply exceeds demand, the excess good can be wasted (or sold for an inefficiently low price); if demand exceeds supply, consumers’ needs go unfulfilled and money goes unspent (and eBay reselling flourishes).
PRICE
Price balances supply and demand. High demand (or low supply) equals a high price: only people with serious levels of demand will be willing to pay enough. Low demand (high supply) means low price—also known as the theory of big-box stores.
“You get what you pay for” is proved false by the idea of price. More apt is “you pay the price the market dictates, eve
n if what you get is a piece of junk.” Your goal as a geeky spendthrift is to avoid popular, scarce junk, while finding unpopular, common treasures.
MARGINALISM
Like the first two ideas, marginalism is another teeter-totter concept that balances markets for consumer and producer.
Consumer: when cost exceeds what something is worth to you, you stop buying it. Producer: when production cost exceeds the amount you can sell something for, you stop producing it. The proverbial profit to be found near the margin refers to the goal of producers to set a price that is near, but just below what people will pay, thus squeezing every red cent out of the consumer.
FIVE LATIN PHRASES TO SHOUT WHILE RIDING INTO BATTLE
Aut vincere aut mori!
To conquer or die!
Per aspera ad astra!
Through difficulties to the stars!
Aut viam inveniam aut faciam!
I'll either find a way or make one!
Fortes fortuna iuvat!
Fortune favors the brave!
Qui audet adipiscitur!
He who dares, wins!
A SAM LOYD PICTURE PUZZLE
Trace and then cut out the six pieces below. Rearrange them to make the best possible representation of a running horse.
SIX BAR BETS YOU CAN WIN
1. HOUSE OF STRAWS
Use six straws to create the classic house shape (a rectangular body with two straws forming the roof, all lying flat on the table). Bet that you can make four equal triangles by moving only three straws. Try it! To all but the most creatively freethinking, this is impossible. The trick is to go 3-D—pick up the three straws that make the bottom and sides of the rectangle and replace them so that one end of each straw is rooted in a corner of the triangle with all three moved straws touching above the center of the original triangle, like a tent or teepee—four equal triangles, each the size of the original roof.
phrase per aspera ad astra is the official motto of the South African Air Force, the city of Gouda in the Netherlands, Kansas, the Royal Air Force, and Pall Mall cigarettes.
2. PAPER MATCH
It is surprising how infrequently this bet's simple trick is discovered. Bet that you can throw a paper match into the air and make it land on its narrow side. This sounds impossible, and it is—until you bend the match!
3. TWO GLASSES I
Submerge two identical glasses in water and place them opening-together, so that when you take them out of the water and set them on the bar it looks as if any bump would separate the glasses, spilling the water out of the top glass. Bet that you can put a dime into the bottom glass without spilling a drop. Impossible, right? Wrong—surface tension makes this very possible. Gently tap the top glass until the glasses separate just enough to slip a dime through the exposed gap into the bottom glass.
4. TRUE MATH GENIUS
This trick will bring a smile to the face of even the most hardened math geek. First, lay matches on a table to form the equation I + II + III = IIII (crossed matches make the plus signs and parallel matches make the equals sign). Challenge your opponent to make this statement true by moving only one match. The trick is to pick up one match from the II, and lay it across the middle match in the III, making the full equation read: I + I + I + I = IIII. Or move a match from II to IIII, making I + I + III = IIIII.
5. TWO GLASSES II
This time, use shot glasses—fill one with colorful liquor and the other with water. Bet that you can make the liquids in the two glasses trade places without using any additional receptacles including your mouth or other glasses. Here's the trick: cover the liquor shot glass with a driver's license or other plastic card and upend it, stacking it face-together with the water shot glass, which sits upright on a table. Gently remove the card—the heavier liquid will flow into the bottom glass, displacing the lighter water into the top glass.
6. BOTTLE CAP T
Arrange six bottle caps on the bar in the shape of a T, with four caps vertically and the remaining two caps on either side of the uppermost cap. Bet that by moving only one cap, you can create two lines of four. Like the first bar bet, the trick is thinking three-dimensionally—take the lowest cap and put it on top of the highest, middle cap. Both the horizontal and the vertical line now contain four caps!
Always the joker, the physicist Richard Feyn-man described leaving a two-nickel tip (generous at the time) with each nickel in an overturned glass full of water (he placed a card over each glass's top, turned over the cup, and then slowly removed the card). After spilling the first glass, how do you think Feynman's regular waitress retrieved the second nickel? In fact, she spilled the second glass, too, then cleaned up the mess and gave Feynman the stink-eye from then on. However, she should have gently slid the glass to the table edge and drained the water into a bowl before retrieving the tip (perhaps then deciding which bodily fluid would best augment Feynman's subsequent dinner orders).
FIVE UNSOLVED PROBLEMS IN MATHEMATICS
1. THE PERFECT CUBOID
A perfect cuboid is one in which the lengths of all edges are integers, the face diagonals are integers, and the body diagonal is also an integer. No example of a perfect cuboid has yet been found, but no one has proven that it can't exist.
2. LYCHREL NUMBERS
Most numbers eventually form a palindrome when the digits are reversed and then added, for example 56 + 65 = 121 and 57 + 75 = 132 + 231 = 363. Numbers that don't (for example, 196 and 879) are Lychrel Numbers. The question: do these numbers really never form palindromes? Can you prove it?
3. ODD, WEIRD NUMBERS
Odd, weird mathematicians have been proven to exist, but odd, weird numbers have not. A weird number is one in which the sum of its divisors is more than the number itself and no combination of its divisors adds up to the number (70: 1 + 2 + 5 + 7 + 14 + 35 = 74). There are even weird numbers aplenty (836, 4030, 5830, etc.), but mathematicians are still searching for the first odd one.
4. GOLBACH'S CONJECTURE
In 1742 the mathematician Christian Golbach hypothesized that every even integer greater than two can be expressed as the sum of two prime numbers (12 = 5 + 7, 14 = 3 + 11, etc.). To date, no one has proven him wrong … or right.
5. HILBERT'S SIXTEENTH PROBLEM
German mathematician David Hilbert's sixteenth problem involves the search for the upper bound of the number of limit cycles in polynomial vector fields. Understanding the previous sentence is the first step toward solving the problem.
THREE MULTIPLICATION TRICKS YOU CAN DO IN YOUR HEAD
MULTIPLY UP TO 20 × 20
For example, take 17 × 13.
Place the larger number on top, in your head.
Imagine a box, encompassing the 17 and the 3.
Add these to make 20.
Add a zero to this, to make it 200.
Multiply the 7 and the 3 to get 21.
Add this to 200 to get the answer: 221.
the pure quest for mathematical knowledge doesn't make you spring to the chalkboard, what about a million bucks? That's the reward offered by the Clay Mathematics Institute for solving any of the Millennium Prize problems: P versus NP, the Hodge conjecture, the Poincaré conjecture, the Reimann hypothesis, the Yang-Mills existence of mass gap, the Navier-Stokes existence of smoothness (mathematical, not social), and the Birch and Swinnerton-Dyer conjecture. In fact, most people consider Grigori Perelman's solution to the Poincaré adequate, but so far Perelman hasn't pursued the million bucks.
MULTIPLY ANY TWO-DIGIT NUMBER BY 11
For example, take 79.
Add a space between the two digits, making 7 9
Add 7 and 9 to get 16.
Put 16 into the space (remember to carry the tens digit!).
The answer is 869.
MULTIPLY BY 9 ON YOUR FINGERS
Lay your spread fingers on a table in front of you.
To multiply by 4, fold down the fourth finger from the left.
Now read your fingers: There are three to the left
of the fold and six to the right—this makes 36.
GET RICH (NOT SO) QUICK: COMPOUND INTEREST CALCULATOR
Compound interest grows exponentially, like little boy and girl bunnies that have vastly increasing numbers of bunnies in successive generations (because their twenty bunny children each have twenty children, etc.). How quickly does your money grow? If you simply stick a chunk of change in ye olde money market fund that compounds continuously and earns 6.5 percent a year, you can use the following equation: P = Cert. P is the future value, C is the initial deposit, e is the very cool transcendental number approximately equal to 2.718, r is the interest rate as a decimal, and t is the number of years it sits. In this scenario, it takes about fifty years for an initial investment of $40,000 to reach a million dollars.