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2010 | Buch

Figuring It Out

Entertaining Encounters with Everyday Math

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This is a book of mathematical stories — funny and puzzling mathematical stories. They tell of villains who try to steal secrets, heroes who encode their messages, and mathematicians who spend years on end searching for the best way to pile oranges.

There are also stories about highway confusions occurring when the rules of Cartesian geometry are ignored, small-change errors due to ignorance of ancient paradoxes, and mistakes in calendars arising from poor numerical approximations.

This book is about the power and beauty of mathematics. It shows mathematics in action, explained in a way that everybody can understand. It is a book for enticing youngsters and inspiring teachers.

Nuno Crato is a leading science writer and mathematician, whose entertaining essays have won a number of international awards.

Inhaltsverzeichnis

Frontmatter

Everyday Matters

Frontmatter
The Dinner Table Algorithm

If you want to invite some friends to a dinner party, but your dining table will only accommodate four people, then you might be faced with a dilemma: how do you choose three compatible dinner companions from among your five closest friends? Your buddy Art has recently broken up with his girlfriend Betty, who is now dating Charlie. Charlie and Art have managed to remain friends, but Charlie is not speaking to Dan, who won’t go anywhere without Eva, who can’t stand Art. So how can you choose your three dinner companions to have a pleasant, hassle-free evening? The best way, believe it or not, would be to make use of an algorithm, which is a set of rules that enable you to search systematically for an answer.

Nuno Crato
Cutting the Christmas Cake

When a small cake has to be cut in two pieces to be shared by two people, and the person who cuts the cake is also the person who chooses which half to take, then there is no guarantee that one of the two people will not be disadvantaged. The best way to avoid any complaints about the division of the cake is for one person to cut the cake and the other to choose which half to take. This way, it is in the first person’s interest to divide the cake as fairly as possible, as otherwise he or she might very well end up with the smaller piece. It is a wise solution, requiring that two persons, basically motivated by egotism, cooperate with one another in such a way that neither is deprived of a fair share.

Nuno Crato
Oranges and Computers

For more than 2000 years mathematics has been making progress by means of rigorous proofs, based on explicit assumptions and logical arguments. The arguments should be faultless. But how can their validity be checked? This has always been the subject of debate and has never been completely resolved. The issue was rekindled at the end of the 20th century, when some prestigious mathematical journals accepted proofs completed with the help of computers. Should these proofs be accepted as legitimate? Should they even be considered mathematical proofs?

Nuno Crato
When Two and Two Don’t Make Four

Two and two always makes four. But the

four

can result from the sum “two plus two” or from the sum “one plus three”. It would seem impossible to differentiate between the two fours. However, this problem has a tremendous practical importance for statistics.

Nuno Crato
Getting More Intelligent Every Day

Ever since intelligence tests were first invented, almost one hundred years ago, there has been a spectacular upsurge in their average results. The increase has been most surprising in the less specific tests, such as the intelligence quotient (IQ) tests, which are supposed to assess various types of intelligence. Could this be true, or are there major errors in the test concepts? This is a difficult question, and psychologists, statisticians and psychometricians do not agree on how to interpret the test results.

Nuno Crato
The Other Lane Always Goes Faster

Jack and Anna leave their respective homes at 8 and have to drive over a bridge to get to the office where they both work. The traffic begins to back up long before they reach the bridge, but each of them handles the situation differently. Whereas Anna remains calm and stays in the right-hand lane, Jack, whose car is behind hers, soon switches over to the left-hand lane and overtakes her. Up ahead, his lane comes to a halt, and Jack is forced to sit there and watch as the cars in the right-hand lane now pass him. Then, taking advantage of a gap, he abruptly decides to rejoin the right-hand lane. A bad decision, as right then his new lane stops again. He waits a short time, frustrated and unable to do anything, until once more a gap opens up in the other lane. He makes use of it to change lanes again in an even riskier maneuver than last time. Now, he feels as if he is gaining ground, until the traffic grinds to a halt yet again. This pattern is repeated over and over. Anna, on the other hand, simply stays put in the right-hand lane. Despite all of his risky lane switches, Jack survives the perils of the road, and eventually gets to work, even arriving on time. He thinks his driving maneuvers have paid off, until he sees that Anna has already parked her car and is walking into the office building.

Nuno Crato
Shoelaces and Neckties

Mathematicians just love problems taken from real life that are easy to formulate. Often they turn out to be the most difficult, and therefore frequently the most interesting. This creates great enthusiasm among them for such apparently trivial questions as finding the best way to lace your shoes!

Nuno Crato
Number Puzzles

The puzzle I am about to describe has an intriguing name, and one you’ve surely heard about. You can write it as “sudoku” or “su doku”, as you prefer. It comes from the Japanese: “su” means “number” or “counting”, and “doku” means “single” or “unique”. In order to complete this puzzle, you have to insert numbers into empty boxes. And there is only one solution.

Nuno Crato
Tossing a Coin

If we toss a coin and it comes up heads, we don’t find this unusual. It is just as likely that it will come up heads as tails. But if it comes up heads five times in a row, we would say that that it is unusual. And what if the order was different? Let’s call heads 1 and tails 0 to make things simpler from now on. For example, if the sequence was 01001, would that also be unusual?

Nuno Crato
The Switch

In a marvelous book written several years ago, Witold Rybczynski wondered which invention would turn out to be the tool of the millennium. After checking out various possible contenders, he chose the screw and the screwdriver. Appropriately enough, his book is called

One Good Turn: A Natural History of the Screwdriver and the Screw.

Nuno Crato
Eubulides, The Heap and The Euro

The euro coins have been in circulation for a few years now, so people in the eurozone should all be able to identify them. Why, then, are there still many people who get confused by them? Some people find it difficult to distinguish a two-cent coin from 5 cents, while others get the 10 and 20 cent coins mixed up, or confuse the 20 cents with the 50 cents. Is that our fault, or is it the design of the coins that is to blame for our confusion?

Nuno Crato

The Earth is Round

Frontmatter
How GPS Works

For thousands of years man navigated by the stars. But since the invention of GPS, we have replaced the Pole Star, the Southern Cross and the Sun with artificial satellites as our primary navigational guides. This may seem far less romantic, but you have to admit that the inner workings of GPS are intriguing. How does it work? Are there satellites watching us from the skies and following our every move, our every position? The short answer is, not really, though the reality, while less frightening, is much more fascinating.

Nuno Crato
Gear Wheels

A person examining the interior of a mechanical clock cannot fail to be amazed at the number of gear wheels it contains. These gear wheels ensure that the clock’s hands revolve at a certain speed by converting the oscillations of the internal energy source, usually at one-second intervals, into a one-hour cycle for the minute hand and a twelve-hour cycle for the hour hand.

Nuno Crato
February 29

February 29 is a date that only comes around every 4 years. If you were born on this date, you know that your birthday only falls in leap years, that is, those years having 366 instead of 365 days, with an extra day in February. Depending on your point of view, that is either an unfortunate stroke of fate, or a reason to celebrate.

Nuno Crato
The Nonius Scale

In the 16th century, sea navigation still depended on mariner’s astrolabes and other relatively primitive instruments for measuring astronomical altitudes. The precision of these instruments was greatly limited by the graduated scale they used, which was normally based on a minimum unit of one degree and could be subdivided into half-degrees but not into smaller units, as the measurement marks engraved in the metal instruments had to have a certain width, and began to become indistinguishable if placed too close together.

Nuno Crato
Pedro Nunes’ Map

When you fly from, say, my home, Lisbon to New York, you usually reach the U.S. coastline at least an hour before your plane lands. During this hour you can typically look down and see the indented outline of the Massachusetts coast and the island of Martha’s Vineyard, Nantucket, and other landmarks. Then you will soon see Long Island and all its beaches, including those of the Hamptons, as you travel northeast to southwest towards New York City. If you were to look at a map of this route, it would seem that the plane had taken a long way round and that, instead of taking the shortest route across the Atlantic, it had reached the coast of the New World farther to the north, where the Portuguese first landed, and then followed the coast.

Nuno Crato
Lighthouse Geometry

Strolling along the coast of the sea on a late summer evening, we can sometimes discern the flash of a lighthouse in the distance, blinking intermittently, as if trying to send us a signal. And indeed it is! The lighthouse is telling us its name. It is sending us a message that is known technically as its

light characteristic

.

Nuno Crato
Asteroids and Least Squares

Eugene Wigner (1902–1995), who was awarded the 1963 Nobel Prize for Physics, wrote an article in 1960 that has since become a classic, titled: “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In Wigner’s words: “There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed an scientific article to his former classmate. The article included, not unusually, the Gaussian distribution. The statistician explained to his former classmate the meaning of the symbols for actual population, for average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. ‘How can you know that?’ was his query. ‘And what is this symbol here?’ ‘Oh’, said the statistician, ‘this is pi.’ ‘What is that?’ ‘The ratio of the circumference of the circle to its diameter.’ ‘Well, now you are pushing your joke too far’, said the classmate, ‘surely the population has nothing to do with the circumference of the circle’”.

Nuno Crato
The Useful Man and the Genius

If you took the time during the 2009 International Year of Astronomy to observe the sun, you would likely have noticed that our star was then without its famous dark spots. This lasted for some time. The sun was spotless on 266 of the 366 days in 2008, and all the way up to October of 2009 there were still virtually no spots to be observed. It is not uncommon for the sun to appear spotless for a brief time, but it is unusual to see it without sunspots for such a prolonged period. That had not occurred since 1913.

Nuno Crato

Secret Affairs

Frontmatter
Alice and Bob1

Alice and Bob live apart from each other and can only communicate by “snail” mail. But they know that the mailman reads all their letters. Alice has a message for Bob and doesn’t want the mailman to read it. What can she do? She has already thought of having the message delivered in a padlocked box. But how can she get the key to Bob? She can’t send it inside the box, because then Bob couldn’t open the box.

Nuno Crato
Inviolate Cybersecrets

Are you apprehensive about sending your credit card number to a website? Is there a CD or book you decided not to buy because the seller required you to use Visa or MasterCard on the Internet? Well, you are certainly not the only person who doesn’t trust the web. Many people around the world still do not take advantage of the efficiencies of e-commerce because they do not have confidence in the security of online transactions. But you might be surprised to find out that sharing confidential information through the web is actually one of the most secure transactions ever devised. If you take some elementary precautions, such as having nothing to do with sellers who are not known to you and not sending confidential information by normal (i.e. non-encrypted) email, the world of e-commerce is at your fingertips.

Nuno Crato
Quantum Cryptography

The integrity of bank transactions, e-commerce and military signals is ensured by the utilization of very secure cryptographic systems. Very secure, however, does not mean absolutely secure. The security of the most reliable modern cryptographic systems, the RSA system, hinges on the difficulty of determining the prime factors of very large numbers.

Nuno Crato
The FBI Wavelet

The language of mathematics can seem esoteric and purely abstract, but many of its constructions end up having surprising applications. One of the most recent and spectacular successes of mathematics is taking place in the processing of signals, and in particular in the processing of images. This new technique has a quaint name: wavelet analysis.

Nuno Crato
The Enigma Machine

Since ancient times men have dreamed of constructing an automatic coding machine. As far as we know, the first attempt was made by the Renaissance architect Leon Battista Alberti (1404–1472), who positioned one concentric disk on top of another, each one inscribed with all the letters of the alphabet. By turning one disk to a certain position with respect to the other, he was able to pair each letter with another, which served to automate the task of encrypting messages by substituting letters. The device may only have mechanically reproduced actions that could be done mentally, but it did ensure that no substitution errors were made in the process.

Nuno Crato

Art and Geometry

Frontmatter
The Vitruvian Man

This is one of the introductory scenes in

The Da Vinci Code.

1

The museum curator Saunière had drawn a circle around himself and painted a five-pointed star in blood on his stomach. In the following 500 pages Professor Langdon will provide an explanation of the geometry of the star and the figure of the Vitruvian Man. The book is a mixture of facts and fiction, which is perfectly acceptable as it is a novel. But as a reader you are entitled to know which of the book’s facts have not been embellished.

Nuno Crato
The Golden Number

There are numbers that surprise us. They pop up unexpectedly in all sorts of situations. For example, take π, the number that represents the quotient of the perimeter of a circumference by its diameter. This number also appears in equations representing the area of a circle, as well as the surface and volume of a sphere. It is not difficult to accept this, as the circumference must have something to do with these other measurements. But it is not so easy to understand the reason why π also appears in statistics, in complex exponential functions and even in the sum of numerical series such as 1 + 1/4 + 1/9 + 1/16 …

Nuno Crato
The Geometry of A4 Paper Sizes

The paper format generally used in photocopiers and printers everywhere outside North America, and which is also generally used for letters and writing pads, has the curious name of A4. Measuring 210 × 297 mm (approximately 8¼ × 11¾ in.), A4 sheets are an unusual size; it would certainly seem more logical if this measurement were a round number. Why not 200 × 300 mm, for example?

Nuno Crato
The Strange Worlds of Escher

Maurits Cornelis Escher was born in 1898, in the city of Leeuwarden in the Netherlands. During his life he produced the most intriguing and mathematically sophisticated woodcuts any artist has ever created.

Nuno Crato
Escher and the Möbius Strip

Escher was once quoted as saying: “In 1960 I was exhorted by an English mathematician (whose name I do not call to mind) to make a print of a Möbius strip. At that time I scarcely knew what it was”.

1

He responded to this challenge by producing two images that became famous:

Möbius Strip I

and

Möbius Strip II

, which I’ve reproduced here. In the first of these woodcuts, which seems to depict three snakes biting each others’ tails, Escher invites us to follow the line of the snakes. What we discover, to our surprise, is that the three reptiles are all on the same surface even though they appear to be following two distinct orbits. In the second woodcut,

Möbius Strip II,

we see nine ants all crawling in the same direction. This time Escher asks us to follow their path and confirm that it is indeed a path without end, because no matter which starting point you choose, you always end up at the same point. The ants appear to be crawling on two separate sides of a single surface, but ultimately each of them travels the entire length of the surface on which they are crawling. In both these images the paths are endless.

Nuno Crato
Picasso, Einstein and the Fourth Dimension

There is an amusing story I have been told about Picasso. When he was already in his sixties, and one of the most famous artists in the world, a very wealthy elderly lady asked him to paint her portrait. The painter did not show any interest whatsoever, but the woman insisted, offering to pay him whatever he wanted. Picasso, fed up with her entreaties, dashed off half a dozen lines on a piece of paper and handed it to the lady. “That will be ten thousand dollars”, he snapped. “Ten thousand?” she asked, astounded. “But you didn’t even take a minute to draw that!” Picasso is said to have retorted: “A minute! You are completely wrong. It took me 60 years.”

Nuno Crato
Pollock’s Fractals

Jackson Pollock (1912–1956) is well known for his gigantic pictures that combine colored lines, splashes of paint, extensive spirals and rhythmical tracks. But he is just as well known for the controversy his art has generated. Some people have asserted that a monkey could paint more interesting pictures than Pollock’s, or have commented that it is impossible to tell the difference between his pictures and completely random scrawls. How could this man have consciously created such strange, chaotic pictures?

Nuno Crato
Voronoi Diagrams

There are mathematical concepts that emerge gradually, springing from within a wide range of contexts that are apparently unconnected and of limited interest. And then, all of a sudden, these same concepts begin to attract the attention of specialists, give rise to numerous studies and applications, and end up contributing to the development of new fields of study. Voronoi diagrams are an example of this type of concept. They were first described systematically in a 1908 article by the mathematician Georgi Voronoi, but their roots can be traced all the way back to ideas first presented in 1644 by René Descartes on the distribution of the planets in the solar system, as well as to work undertaken in the middle of the 19th century by the German mathematician G. L. Dirichlet.

Nuno Crato
The Platonic Solids

Have you ever looked closely at a cube? It is one of the commonest solids. In nature it appears in crystals; in our homes in furniture design; in casinos, cubes can be seen in the dice rolled on betting tables.

Nuno Crato
Pythagorean Mosquitoes

Before people have regular exposure to other cultures, there is a natural tendency to assume that everything that happens in the world mirrors our own experiences. We might think, for example, that everyone everywhere else eats a fried breakfast, or that there is no city on earth more hospitable than our own. But then one day we encounter sushi, or read the work of writer Machado de Assis, and begin to see that there are many ways of living life that are different from our own way.

Nuno Crato
The Most Beautiful of All

How can we find beauty in an equation? Readers will certainly have divided opinions about this. Some people will assume the question is ironic: what possible beauty could there be in those incomprehensible squiggles that filled our schoolbooks? But in the view of others who carried on with math after leaving school, and came to enjoy this subject, even making it their life’s work, the simplicity and elegance of certain equations make them beautiful. Quite beautiful, in fact, though in some cases it is difficult to explain the reasons for their beauty. One is surely the strange condensation of reality they conveyed, reality that may be geometrical, physical, biological, or purely ideal. And then there is their flexibility, their applicability to infinite numbers of unexpected situations, as well as their graphic representation.

Nuno Crato

Mathematical Objects

Frontmatter
The Power of Math

“How can it be” wondered Einstein, “that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?”

1

Nuno Crato
Doubts in the Realm of Certainty

In the early 1980s Philip Davis and Reuben Hersh wrote a bestselling book entitled

The Mathematical Experience

. The book’s philosophical message is found throughout the narrative of the intellectual adventure that has given rise to modern mathematics. In another book by Hersh published in 1997, called

What is Mathematics, Really?

, the author makes his philosophical leanings clear. It was enough to re-ignite a lively debate on the fundamental aspects of this mysterious entity that is mathematics.

Nuno Crato
When Chance Enhances Reliability

Randomized algorithms have revolutionized the way mathematics operates. Surprisingly, making use of chance may be the fastest way to obtain the solution to a problem, and not only when it comes to math.

Nuno Crato
The Difficulty of Chance

At first glance nothing would seem to be as natural and easy as chance. It is orderliness and organization that seem difficult to achieve. But reality is very different. Things frequently take on lives of their own, organizing themselves and creating patterns. In a forest, for example, each type of tree tends to appear in specific areas, simply because the greatest concentration of the seeds of that tree end up in those areas. In the oceans, waves seem to move in concert, in groups, in consecutive lines, because the water masses move in coordination. In the universe, the stars appear to be grouped into galaxies, which in turn gather in groups, and then those groups cluster as superconglomerates.

Nuno Crato
Conjectures and Proofs

The so-called Collatz conjecture was formulated in 1937 by the German mathematician Lothar Collatz. A conjecture concerns a mathematical assumption, something that we think is true, but that has never been finally proved or disproved. Like some of the most famous mathematical assumptions, these conjectures are easy to understand and they pass the common sense test, but they are tremendously difficult to prove or disprove.

Nuno Crato
Mr. Benford

Would you like to win a bet? Let’s see if you can manage to guess the first digit of a number or at least whether that number is less than 4. Tell somebody to think of everyday-life numbers, such as the street number of a house or the amount of money they have in their checking account, or a percentage rate they commonly use such as a mortgage interest rate, or, perhaps, a physical or mathematical constant such as pi. If your bet is that this number begins with 1, 2, or 3, you will be right more than half the time, in fact, you will be right 60.2% of the time. Your betting partner will no doubt be surprised by your success rate, for at first sight there is no reason at all for the laws of chance to produce such a result.

Nuno Crato
Financial Fractals

The mathematician Benoît Mandelbrot once said that his childhood ambition was to become the “Newton” of a specific mathematical field, no matter how small, in other words to propose something creative and completely innovative in a particular area of mathematics. Ultimately he chose what came to be called fractal objects, at that time relatively unknown geometrical figures with curious properties.

Nuno Crato
Turing’s Test

Will we succeed one day in constructing machines capable of thinking? This is a more profound question than it at first seems, and has been one of the most hotly debated topics among philosophers over the centuries. When computers first appeared, these debates flared up again and their tone became more pressing. Nowadays, with computer programs capable of beating the world’s chess champion, it makes a lot of sense to ask whether, when all is said and done, artificial intelligence is not in fact true intelligence.

Nuno Crato
DNA Computers

On April 25, 1953, James D. Watson and Francis Crick published an article in the journal

Nature

that was less than two pages long. In this study they presented the famous double-helix structure of DNA, and as a consequence the world today has changed. The discovery of DNA has opened up new paths in biology, medicine, agriculture, forensic science, and in numerous other technical and scientific fields. The double helix has appeared in works of art, and the abbreviation has even been adopted as the name for perfumes. And now something that people have been talking about for some time seems to be coming true: DNA could begin to be used for computation. In the near future PCs might no longer use silicone, but instead an aqueous solution of DNA molecules.

Nuno Crato
Magical Multiplication

The video shows what it claims is a new method of multiplying. It is simple, somewhat strange, but it always seems to work. It starts by showing us how to multiply 21 by 13. To do this we have to draw two horizontal lines, which represent the number of tens (2) in the first number. Below we draw a single line that represents the number of units (1) it contains. Then we draw the second number using vertical lines: one on the left for the number of tens (1), and three on the right for the number of units (3).

Nuno Crato
π Day

We all know certain commemorative dates by heart. In the U.S., for example, Mother’s Day falls on the second Sunday in May, Father’s Day on the third Sunday in June, Labor Day on the first Monday in September. And π day? Do you know when that is?

Nuno Crato
The Best Job in the World

Many scientists will secretly admit that they enjoy their jobs so much they would pay to do what they are paid for. Of course, this is something of an exaggeration. Very few of them could actually keep up a full-time job in a scientific field if they were not paid. The “good old days” when aristocrats pursued science as a hobby are long past. Today, science is a job. A good job.

Nuno Crato

Out of this World

Frontmatter
Electoral Paradoxes

Voting in elections is one of humanity’s great achievements. No better method has yet been invented to achieve a system of government that guarantees liberty and progress. But would it be possible to invent a better one?

Nuno Crato
The Melon Paradox

This is a curious problem that comes up regularly in math competitions. You will see it in one form or another in published collections of problems from these competitions. It’s not that the math itself is difficult. The difficulty lies in believing the results. As an example, let’s start with a melon weighing 50 ounces. Only 1% of the mass of the melon is made up of solid matter, while the remaining 99% is water. The melon is left in the sun and dehydrates to such an extent that it now only contains 98% water. The question is: how much does the melon weigh now? The answer is easy, provided you do your sums properly. But let’s start by guessing the weight. Will it be about 49 ounces? Or even 49 and a half ounces? Or just 45 ounces?

Nuno Crato
The Cupcake Paradox

When a fair-minded group of friends shares a plate of cupcakes, each one takes one and eats it, taking care to leave a cupcake for the next person. However, if the cupcakes are especially tasty and everyone is hungrier than usual, what happens when there is only one cupcake left on the plate? Carefully, one of the friends cuts the cake in half and takes one of the halves. A second person can’t resist, and cuts the remaining half in half. Then a third friend comes forward and cuts the remainder in half. And so it goes on … Theoretically we could imagine a virtual cake that is infinitely divisible, and a group of friends with all the time in the world to go on eating half of whatever was left of the cake.

Nuno Crato
Infinity

Galileo, whose scientific activities were celebrated during the International Year of Astronomy, considered various paradoxes having to do with infinity. One of the simplest and most illustrative paradoxes concerns two sets, one of the natural numbers (1, 2, 3, …), and one of their doubles (2, 4, 6, …). We can establish a one-to-one (bijection) correspondence between the two sets: 1 corresponds to 2, 2 corresponds to 4, 3 corresponds to 6, and so on. The first set seems to contain twice as many elements as the second set, because it contains both odd and even numbers. But doesn’t the fact that we can establish a one-to-one correspondence between each number and its double indicate that each set has the same number of elements?

Nuno Crato
Unfair Games

Imagine that we are in a casino that is promoting the following game: We put 100 dollars on the table, and win or lose by tossing a coin. If it is heads, we win 40 dollars, and if it is tails we lose 30 dollars. Should we join the game?

Nuno Crato
Monsieur Bertrand

We expect to receive two Olympic medals, and we know that neither of them is bronze. There are three boxes in front of us, each containing two medals. One contains two gold medals (GG), another two silver medals (SS), and the third one gold and one silver medal (GS). The boxes are indistinguishable from one another, each with two drawers containing one medal. This is all the information we know. We select a box at random, open one of the drawers and find a silver medal inside. What is the probability that there will be a gold medal in the other drawer of this box? That seems easy. We have eliminated the possibility of the box containing two gold medals (GG), so there are two hypotheses: we selected the box with two silver medals (SS) or the box with one gold and one silver medal (GS). That seems to be it – the probability of finding gold in the other drawer of the box is 1 in 2.

Nuno Crato
Boy or Girl?

Mary and John have two children. Their first-born is a boy named Jack. What is the probability that the couple have two children of different genders? This seems at first to be a ridiculously simple question. If we concede that it is just as probable for a boy or a girl to be born, and if we also concede that this has nothing to do with the gender of the first-born baby, then there is no doubt that the probability that the second child will be a girl (therefore not the same gender as the first-born) is 1 in 2. And that is the answer: 1 in 2.

Nuno Crato
A Puzzle for Christmas

Everyone knows that Santa Claus likes to please people. But he doesn’t like to waste presents. He put money in my stocking. But we came to an agreement, he and I. Or rather, he explained the rules of the game to me.

Nuno Crato
Crisis Time for Easter Eggs

Every year the Easter Bunny has an infinite number of eggs available for distribution. Nobody knows where he gets them from, or how he manages to get them to all the children on Easter morning.

Nuno Crato
Backmatter
Metadaten
Titel
Figuring It Out
verfasst von
Nuno Crato
Copyright-Jahr
2010
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-04833-3
Print ISBN
978-3-642-04832-6
DOI
https://doi.org/10.1007/978-3-642-04833-3