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Impossible?: Surprising Solutions to Counterintuitive Conundrums

Julian Havil
Copyright Date: 2008
Pages: 264
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  • Book Info
    Book Description:

    InNonplussed!, popular-math writer Julian Havil delighted readers with a mind-boggling array of implausible yet true mathematical paradoxes. Now Havil is back withImpossible?, another marvelous medley of the utterly confusing, profound, and unbelievable--and all of it mathematically irrefutable.

    Whenever Forty-second Street in New York is temporarily closed, traffic doesn't gridlock but flows more smoothly--why is that? Or consider that cities that build new roads can experience dramatic increases in traffic congestion--how is this possible? What does the game showLet's Make A Dealreveal about the unexpected hazards of decision-making? What can the game of cricket teach us about the surprising behavior of the law of averages? These are some of the counterintuitive mathematical occurrences that readers encounter inImpossible?

    Havil ventures further than ever into territory where intuition can lead one astray. He gathers entertaining problems from probability and statistics along with an eclectic variety of conundrums and puzzlers from other areas of mathematics, including classics of abstract math like the Banach-Tarski paradox. These problems range in difficulty from easy to highly challenging, yet they can be tackled by anyone with a background in calculus. And the fascinating history and personalities associated with many of the problems are included with their mathematical proofs.Impossible?will delight anyone who wants to have their reason thoroughly confounded in the most astonishing and unpredictable ways.

    eISBN: 978-1-4008-2967-5
    Subjects: Statistics, Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-x)
  3. Acknowledgments
    (pp. xi-xiv)
  4. Introduction
    (pp. 1-2)

    We begin with a classic puzzle.

    Imagine a rope just long enough to wrap tightly around the equator of a perfectly spherical Earth. Now imagine that the length of the rope is increased by 1 metre and again wrapped around the Earth, supported in a regular way so that it forms an annulus. What is the size of the gap formed between the Earth and the extended rope?

    The vast Earth, the tiny 1 metre—surely the rope will be in effect as tight after its extension as before it? Yet, let us perform a small calculation: in standard notation,...

    (pp. 3-10)

    Part of Stevie Nicks’s 2003 lyrics of the Fleetwood Mac song ‘Everybody Finds Out’ might describe the reader’s reaction to the subject matter of this first chapter:

    I know you don’t agree…

    Well, I know you don’t agree.

    The song’s title also finds its way into the title of an episode of the NBC sitcom television seriesFriends, ‘The One Where Everybody Finds Out’, which was first aired in February 1999 and which contains the following dialogue:

    Rachel: Phoebe just found out about Monica and Chandler.

    Joey: You mean how they’re friends and nothing more? [Glares at Rachel]

    Rachel: No....

  6. Chapter 2 SIMPSON’S PARADOX
    (pp. 11-20)

    It is difficult for the non-cricket fanatic to appreciate the trauma associated with the biannual cricket competition between the arch-rivals England and Australia, universally known as theAshes. On 29 August 1882 (at home) a full-strength England cricket side was for the first time beaten by Australia, which caused the British publicationThe Sporting Timesto run an obituary for English cricket which included the words ‘The body will be cremated and the Ashes taken to Australia’. On the return fixture (in Australia) England regained the upper hand and a small urn was presented to the captain, Lord Darnley, in...

    (pp. 21-30)

    The Dutch mathematician, mathematical historian and educator Hans Freudenthal was an original and inspirational thinker. Radio telescopes are pretty complicated mechanisms. Freudenthal therefore (and reasonably) argued that electronic communication with extraterrestrial life would require of them the capacity to count and to recognize that 2 + 2 = 4 and from this conviction he created a mathematically based interstellar language called Lingua Cosmica (thelanguage of the cosmos, which was published in his bookLINCOS: Design of a Language for Cosmic Intercoursein 1960). It seems that he also created a remarkable logical puzzle, which we will consider in this...

  8. Chapter 4 BRAESS’S PARADOX
    (pp. 31-38)

    In the 1969 publication ‘Graphentheoretische Methoden und ihre Anwendungen’, W. Knödel remarked that

    …major road investment in Stuttgart’s city centre, in the vicinity of the Schlossplatz, failed to yield the benefits expected. They were only obtained when a cross street, the lower part of Königstrasse, was subsequently withdrawn from traffic use....

    Eliminating a road, rather than building a road, improved traffic flow.

    Figure 4.1 shows a map of the relevant part of the city, with part of Königstrasse now a pedestrian precinct.

    When 42nd Street in New York City was temporarily closed to traffic, rather than the expected gridlock resulting,...

    (pp. 39-49)

    Apart from admiration for the committed effort and ingenuity displayed in a note to theAmerican Mathematical Monthlyby H. S. Uhler, the casual reader might be surprised by the approximate value

    0.207 879 576 350 761 908 546 955 619 834 978 770 033 877 841 631 769 614

    the author gives to${\text{i}}^{\text{i}} = \sqrt{- 1} ^{\sqrt{- 1}}$: a real number.¹ The purpose of the note was to give high-order decimal approximations to eight numbers, each of which is a power of e; two of those numbers were eπ, the Gelfond constant, which we will mention later, and e−π/2, which was written...

  10. Chapter 6 BUCKING THE ODDS
    (pp. 50-67)

    In this chapter we will consider two problems, each of which caused large-scale consternation and disbelief when they came to the attention of the public. The first had academic origins, the second was inspired by a popular American television show.

    We have already considered matters relating to red- and bluehat wearers not knowing the colour of the hat each is wearing. Chapter 1 had a group of them sitting listening to the chiming of a clock, waiting for revelation. Here we will give each of them a more active role: guessing the colour of their hat, but under the following...

  11. Chapter 7 CANTOR’S PARADISE
    (pp. 68-81)

    Jane Muir began the final chapter of her delightfully written bookOf Men and Numberswith typically elegant prose:

    There are times in history—the history of a man as well as a civilization—when one can look back and say, ‘So this is where it’s all been leading. It seems so obvious now, why didn’t I realize before?’ A man or a civilization comes to the end of a road; the journey is over; all the wanderings and travels down dead ends and over highways have led to this particular place and suddenly he realizes that he is at...

    (pp. 82-87)

    If we work on the middle floor of a building with one elevator and assume that floor usage is uniform, symmetry dictates that, if the elevator is not stationary on our floor, it will arrive at our floor with a probability$\frac{1}{2}$of going up or down. Similar reasoning was used by George Gamow and Marvin Stern when, in 1956, they worked in a building of seven floors, with lowest floor numbered 1 and the highest numbered 7; Gamow’s office was on the second floor and Stern’s the sixth. Whenever Gamow decided to visit Stern the elevator almost always appeared...

  13. Chapter 9 THE TOSS OF A COIN
    (pp. 88-102)

    If the reader was asked to judge whether the following 1679 bits of binary data is random or could possibly contain a message, the answer would most probably be the latter: the eye discerns some sort of ‘lack of randomness’ with those sequences of 0s surely too long for mere chance to have created them. Think of tossing a coin 1679 times and writing a 0 if a head appears uppermost and a 1 otherwise: such long runs of heads would surely cause us to suspect the coin’s fairness. If the reader agrees with this, he or she will be...

  14. Chapter 10 WILD-CARD POKER
    (pp. 103-112)

    Since its invention somewhere in the Louisiana Territory around 1800, poker has evolved into a vastly complex contest of skill and chance. Acknowledged experts spread across the globe, even the late and acclaimed bridge expert, Terence Reese, coauthored a book entitledPoker: Game of Skill. The game exists in many variants, which add novelty and subtlety to the standard rules, but whatever the details of the particular variant there is a ranking of the winning hands. To begin with we list the standard hands, in the order of their rankings (see also table 10.1):

    Straight flush: all five cards in...

  15. Chapter 11 TWO SERIES
    (pp. 113-130)

    In chapter 8 ofNonplussed!we considered the paradoxical solid known (in particular) asTorricelli’s Trumpet. Figure 11.1 shows what this remarkable object looks like. It is formed as the solid of revolution of the curvey= 1/xfor (say)x≥ 1 and calculus was used to show that the trumpet had a finite volume but infinite surface area, the required computations being as follows: the volume is\[ \begin{align*} \lim\limits_{N \rightarrow \infty} \pi \int_1^N \left(\frac{1}{x}\right)^2\ \rm{d}x & = \mathop{\lim}\limits_{N\rightarrow \infty} \pi \int_1^N \frac{1}{x^2}\,\rm{d}x\\ & = \mathop{\lim}\limits_{N \rightarrow \infty} \pi \left[ - \frac{1}{x}\right]_1^N \\ & = \mathop{\lim}\limits_{N \rightarrow \infty} \pi \left(1 - \frac{1}{N}\right)\\ & = \pi \\ \end{align*} \]and the surface area is\[\mathop {\lim }\limits_{N \to \infty } \int_1^N {\frac{1} {x}\sqrt {1 + \frac{1} {{x^4 }}{\text{d}}x} } = \mathop {\lim }\limits_{N \to \infty } \int_1^N {\frac{{\sqrt {x^4 + 1} }} {{x^3 }}} {\text{d}}x = \mathop {\lim }\limits_{N \to \infty } \left\{ { - \frac{1} {{2N^2 }}\sqrt {N^4 + 1} + \frac{1} {2}\ln (N^2 + \sqrt {N^4 + 1} ) + \frac{{\sqrt 2 }} {2} - \frac{1} {2}\ln (1 + \sqrt 2 } \right\}\], which does not exist, since the second term is unbounded for largeN.

    The first calculation is easy and the second comparatively...

  16. Chapter 12 TWO CARD TRICKS
    (pp. 131-145)

    The bewilderment that accompanies a well-performed, good card trick relies greatly on the expertise of the conjurer—the ability to misdirect, to manipulate both cards and observer—and sometimes an underlying principle which is surprising in itself. We are interested in two such principles, both of which are the basis of numerous effects and both of which were discovered by academics.

    The second chapter of Genesis continues the story of the creation of Heaven and Earth and begins (King James Version):

    Thus the heavens and the earth were finished and all the host of them And on the seventh day...

  17. Chapter 13 THE SPIN OF A NEEDLE
    (pp. 146-164)

    InNonplussed!we considered some consequences of tossing a needle, in particular, onto a set of equally spaced parallel lines. The fact, surprising and historically significant, that the probability of the needle landing across a line involved π was first investigated by the eighteenth-century French scientist Georges Louis Leclerc, Comte de Buffon: hence the nameBuffon’s Needle. Here we move to the twentieth century to consider a simple question about spinning the needle, which has its own intriguing and important answer.

    In 1922–23 the eminent American mathematician George Birkoff (and father of Garrett) gave a series of lectures on...

  18. Chapter 14 THE BEST CHOICE
    (pp. 165-175)

    The German mathematician and astronomer Johannes Kepler was married twice, once in 1597 to Barbara Muhleck and, after her death from cholera, to Susanna Reuttinger in 1613: the first marriage was arranged by friends and matchmakers, the second by Kepler himself, who had assessed eleven possible candidates. Since he pondered the relative merits of the individuals for nearly two years, weighing parental standing, dowry size and conflicting advice each against the other he must be judged as a careful suitor. Eventually his decision was made and he explained his ultimate choice to one Baron Strahlendorf in a letter of 23...

  19. Chapter 15 THE POWER OF POWERS
    (pp. 176-189)

    Given its importance to the chapter, ‘Some consequences of the irrationality of log102’ could have been a reasonable alternative title. Yet the title stands, as the subject matter that follows concentrates on some rather surprising consequences relating to the decimal expansion of 2n; we prove the elementary result that log102 is indeed irrational in the appendix (page 226), expending our efforts over the next few pages appealing to that result.

    In a note to the journalMathematics of Computation(E. and U. Karst, The first power of 2 with 8 consecutive zeroes, July 1964, 18(87):508) the authors provided...

  20. Chapter 16 BENFORD’S LAW
    (pp. 190-200)

    At the end of the previous chapter, with the use of the potent Weyl Equidistribution Theorem, we saw that the first digit of powers of 2 are not distributed uniformly over {1, 2, 3, . . . , 9} but rather according to the law$P[{\text{First digit of }}2^n = d] = \log _{10} \left( {1 + \frac{1}{d}} \right)$.

    Moreover, we argued that the phenomenon exists equally with 2 replaced by any number that is not a rational power of 10. Perhaps this behaviour is then a property of powers of integers, but then consider the consumption (measured in kilowatt hours) of the 1243 users of electricity of Honiara in the British...

    (pp. 201-209)

    Exponential notation deals with very big numbers very efficiently. For example,$2^{2^{22} } $has about one million digits in it (and happens to be the biggest number which can be manufactured from four 2s using the standard arithmetic operations). In Adams’s original book of theHitchhikerseries,The Hitchhiker’s Guide to the Galaxy, appears what might be the biggest number ever used in a work of fiction: 2260 199, the quoted odds against Arthur Dent and Ford Prefect being rescued by a passing spaceship, having been thrown out of an airlock. (In fact, they were rescued by a spaceship—powered by...

    (pp. 210-216)

    The subject matter of this last chapter simply has to rank as the most counterintuitive result in mathematics and is a fitting finalé to a book devoted to mathematical surprise.

    Stefan Banach and Alfred Tarski brought to the world an improvement on a paradox devised by the great topologist Felix Hausdorff, the formalized form of which is often replaced by something fanciful such as:

    A solid sphere can be dissected into five pieces and the pieces reassembled to form two complete spheres of exactly the same size as the original.

    Or, alternatively:

    A solid sphere the size of a pea...

  23. The Motifs
    (pp. 217-220)

    The design of the chapter motifs did not demand Carroll’s fantastic, mutually perpendicular brooks and hedges but a pair of infinite invisible lines, intersecting at an arbitrary angle, and repeated at regular intervals. By this means the plane is divided into an infinite number of congruent parallelograms, the vertices of which form an infinite, regular lattice. The two lines determine two independent directions of translation and the lengths of the sides of the parallelograms the fundamental translation distances, which must be bounded below by a number ε > 0. It is with these two independent translations that the study of the wallpaper...

  24. Appendix
    (pp. 221-232)
  25. Index
    (pp. 233-235)