Ian stewart, p.1

Ian Stewart, page 1


Ian Stewart

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Ian Stewart

  Table of Contents

  By the Same Author

  Title Page


  Second Drawer Down . . .

  Calculator Curiosity 1

  Year Turned Upside Down

  Luckless Lovelorn Lilavati

  Sixteen Matches

  Swallowing Elephants

  Magic Circle



  Secrets of the Abacus

  Redbeard’s Treasure


  Who Invented the Equals Sign?

  Stars and Snips

  By the Numbers of Babylon

  Magic Hexagons

  The Collatz-Syracuse-Ulam Problem

  The Jeweller’s Dilemma

  What Seamus Didn’t Know

  Why Toast Always Falls Buttered-Side Down

  The Buttered Cat Paradox

  Lincoln’s Dog

  Whodunni’s Dice

  A Flexible Polyhedron

  But What About Concertinas?

  The Bellows Conjecture

  Digital Cubes

  Nothing Which Appeals Much to a Mathematician

  What Is the Area of an Ostrich Egg?

  Order into Chaos

  Big Numbers

  The Drowning Mathematician

  Mathematical Pirates

  The Hairy Ball Theorem

  Cups and Downs

  Secret Codes

  When 2 + 2 = 0

  Secret Codes That Can Be Made Public

  Calendar Magic

  Mathematical Cats

  The Rule of Eleven

  Digital Multiplication

  Common Knowledge

  Pickled Onion Puzzle

  Guess the Card

  And Now with a Complete Pack

  Halloween = Christmas

  Egyptian Fractions

  The Greedy Algorithm

  How to Move a Table

  Rectangling the Square

  Newton, by Byron

  X Marks the Spot

  Whatever’s the Antimatter?

  How to See Inside Things

  Mathematicians Musing About Mathematics

  Wittgenstein’s Sheep

  Leaning Tower of Pizza

  PieThagoras’s World-Famous Mince πs

  Diamond Frame

  Pour Relations

  Alexander’s Horned Sphere

  The Sacred Principle of Mat

  Perfectly Abundantly Amicably Deficient

  Target Practice

  Just a Phase I’m Going Through

  Proof Techniques

  Second Thoughts

  How Dudeney Cooked Loyd

  Cooking with Water

  Celestial Resonance

  Calculator Curiosity 2

  Which is Bigger?

  Sums That Go On For Ever

  The Most Outrageous Proof

  Colorado Smith and the Solar Temple

  Why Can’t I Add Fractions Like I Multiply Them?

  Farey, Farey, Quite Contrary

  Pooling Resources

  Welcome to the Rep-Tile House

  Cooking on a Torus

  The Catalan Conjecture

  The Origin of the Square Root Symbol

  Please Bear with Me

  The Ham Sandwich Theorem

  Cricket on Grumpius

  The Man Who Loved Only Numbers

  The Missing Piece

  The Other Coconut

  What Does Zeno?

  Pieces of Five

  Pi in the Sky

  The Curious Incident of the Dog

  Mathematics Made Difficult

  A Weird Fact about Egyptian Fractions

  A Four Colour Theorem

  Serpent of Perpetual Darkness

  What Are the Odds?

  A Potted History of Mathematics

  The Shortest Mathematical Joke Ever

  Global Warming Swindle

  Name the Cards

  What Is Point Nine Recurring?

  Ghost of a Departed Quantity

  Nice Little Earner

  A Puzzle for Leonardo

  Congruent Numbers

  Present-Minded Somewhere Else

  It’s About Time

  Do I Avoid Kangaroos?

  The Klein Bottle

  Accounting the Digits

  Multiplying with Sticks

  As Long as I Gaze on Laplacian Sunrise

  Another Take on Mathematical Cats

  Bordered Prime Magic Square

  The Green-Tao Theorem

  Peaucellier’s Linkage

  A Better Approximation to π

  Strictly for Calculus Buffs

  The Statue of Pallas Athene

  Calculator Curiosity 3

  Completing the Square

  The Look and Say Sequence

  Non-Mathematicians Musing About Mathematics

  Euler’s Conjecture

  The Millionth Digit

  Piratical Pathways

  Trains That Pass in the Siding

  Please Make Yourself Clear

  Squares, Lists and Digital Sums

  Hilbert’s Hit-List

  Match Trick

  Which Hospital Should Close?

  How to Turn a Sphere Inside Out

  A Piece of String Walked into a Bar . . .

  Slicing the Cake

  The Origin of the Symbol for Pi

  Hall of Mirrors

  Greek and Trojan Asteroids

  Sliding Coins

  Beat That!

  Euclid’s Puzzle

  The Infinite Monkey Theorem

  Monkeys Against Evolution

  Universal Letter of Reference

  Snakes and Adders

  Powerful Crossnumber

  Magic Handkerchiefs

  A Bluffer’s Guide to Symmetry

  Digital Century Revisited

  An Infinity of Primes

  A Century in Fractions

  Ah, That Explains It . . .

  Life, Recursion and Everything

  False, Not Stated, Not Proved

  Proof That 2 + 2 = 4

  Slicing the Doughnut

  The Kissing Number

  Tippe Top Twister

  When Is a Knot Not Knotted?

  The Origin of the Factorial Symbol

  Juniper Green

  Mathematical Metajoke

  Beyond the Fourth Dimension

  Slade’s Braid

  Avoiding the Neighbours

  Career Move

  A Rolling Wheel Gathers No Speed

  Point Placement Problem

  Chess in Flatland

  The Infinite Lottery

  Ships That Pass ...

  The Largest Number Is Forty-Two

  A Future History of Mathematics

  Professor Stewart’s Superlative Storehouse of Sneaky Solutions and Stimulating Supplements

  Copyright Page

  By the Same Author

  Concepts of Modern Mathematics

  Game, Set, and Math

  Does God Play Dice?

  Another Fine Math You’ve Got Me Into

  Fearful Symmetry (with Martin Golubitsky)

  Nature’s Numbers

  From Here to Infinity

  The Magical Maze

  Life’s Other Secret


  What Shape Is a Snowflake?

  The Annotated Flatland (with Edwin A. Abbott)

  Math Hysteria

  The Mayor of Uglyville’s Dilemma

  Letters to a Young Mathematician

  How to Cut a Cake

  Why Beauty Is Truth

  Taming the Infinite

  Professor Stewart’s Cabinet
of Mathematical Curiosities

  with Jack Cohen

  The Collapse of Chaos

  Figments of Reality

  What Does a Martian Look Like?

  Wheelers (science fiction)

  Heaven (science fiction)

  with Terry Pratchett and Jack Cohen

  The Science of Discworld

  The Science of Discworld II: The Globe

  The Science of Discworld III: Darwin’s Watch


  The following figures are reproduced with the permission of the named copyright holders:

  Pages 30, 280 (‘What Seamus Didn’t Know’); Suppiya Siranan.

  Page 41 (‘What is the Area of an Ostrich Egg?’); Hierakonpolis expedition, leader Renée Friedman, photograph by James Rossiter.

  Page 69 (‘Mathematical Cats’); Dr Sergey P. Kuznetsov, Laboratory of Theoretical Nonlinear Dynamics, SB IRE RAS.

  Page 92 (‘How to See Inside Things’); Brad Petersen.

  Page 107 (‘Alexander’s Horned Sphere’); from Topology by John G. Hocking and Gail S. Young, Addison-Wesley, 1961.

  Page 113 (‘Just a Phase I’m Going Through’); GNU Free Documentation License, Free Software Foundation (www.gnu.org/copyleft/fdl.html).

  Page 182 (‘The Klein Bottle’); Janet Chao (www.illustrationideas.com).

  Page 184 (‘The Klein Bottle’); Konrad Polthier, Free University of Berlin.

  Page 190 (‘Multiplying with Sticks’); Eric Marcotte PhD (www.sliderule.ca).

  Page 216 (‘How to Turn a Sphere Inside Out’); Bruce Puckett.

  Second Drawer Down . . .

  When I was fourteen, I started collecting mathematical curiosities. I’ve been doing that for nearly fifty years now, and the collection has outgrown the original notebook. So when my publisher suggested putting together a mathematical miscellany, there was no shortage of material. The result was Professor Stewart’s Cabinet of Mathematical Curiosities.

  Cabinet was published in 2008, and, as Christmas loomed, it began to defy the law of gravity. Or perhaps to obey the law of levity. Anyway, by Boxing Day it had risen to number 16 in a well-known national bestseller list, and by late January it had peaked at number 6. A mathematics book was sharing company with Stephenie Meyer, Barack Obama, Jamie Oliver and Paul McKenna.

  This was, of course, completely impossible: everyone knows that there aren’t that many people interested in mathematics. Either my relatives were buying a huge number of copies, or the conventional wisdom needed a rethink. So then I got an email from my publisher asking whether there might be any prospect of a sequel, and I thought, ‘My suddenly famous Cabinet is still bursting at the seams with goodies, so why not?’ Professor Stewart’s Hoard of Mathematical Treasures duly emerged from darkened drawers into the bright light of day.

  It’s just what you need to while away the hours on your desert island. Like its predecessor, you can dip in anywhere. In fact, you could shuffle both books together, and still dip in anywhere. A miscellany, I have said before and stoutly maintain, should be miscellaneous. It need not stick to any fixed logical order. In fact, it shouldn’t, if only because there isn’t one. If I want to sandwich a puzzle allegedly invented by Euclid between a story about Scandinavian kings playing dice for the ownership of an island and a calculation of how likely it is for monkeys to randomly type the complete works of Shakespeare, then why not?

  We live in a world where finding time to work systematically through a long and complicated argument or discussion gets ever more difficult. That’s still the best way to stay properly informed - I’m not decrying it. I even try it myself when the world lets me. But when the scholarly method won’t work, there’s an alternative, one that requires only a few minutes here and there. Apparently quite a lot of you find that to your taste, so here we go again. As one radio interviewer remarked about Cabinet (sympathetically, I believe), ‘I suppose it’s the ideal toilet book.’ Now, Avril and I actually go out of our way not to leave books in the loo for visitors to read, because we don’t want to have to bang on the door at 1 a.m. to remove a guest who has found War and Peace unexpectedly gripping. And we don’t want to risk getting stuck in there ourselves.

  But there you go. The interviewer was right. And like its predecessor, Hoard is just the kind of book to take on a train, or a plane, or a beach. Or to sample at random over Boxing Day, in between watching the sports channels and the soaps. Or whatever it is that grabs you.

  Hoard is supposed to be fun, not work. It isn’t an exam, there is no national curriculum, there are no boxes to tick. You don’t need to prepare yourself. Just dive in.

  A few items do fit naturally into a coherent sequence, so I’ve put those next to each other, and earlier items do sometimes shed light on later ones. So, if you come across terms that aren’t being explained, then probably I discussed them in an earlier item. Unless I didn’t think they needed explanation, or forgot. Thumb quickly through the earlier pages seeking insight. If you’re lucky, you may even find it.

  A page from my first notebook of mathematical curiosities.

  When I was rummaging through the Cabinet’s drawers, choosing new items for my Hoard, I privately classified its contents into categories: puzzle, game, buzzword, squib, FAQ, anecdote, infodump, joke, gosh-wow, factoid, curio, paradox, folklore, arcana, and so on. There were subdivisions of puzzles (traditional, logic, geometrical, numerical, etc.) and a lot of the categories overlapped. I did think about attaching symbols to tell you which item is what, but there would be too many symbols. A few pointers, though, may help.

  The puzzles can be distinguished from most other things because they end with Answer on page XXX. A few puzzles are harder than the rest, but none outlandishly so. The answer is often worth reading even if - especially if - you don’t tackle the problem. But you will appreciate the answer better if you have a go at the question, however quickly you give up. Some puzzles are embedded in longer stories; this does not imply that the puzzle is hard, just that I like telling stories.

  Almost all the topics are accessible to anyone who did a bit of maths at school and still has some interest in it. The FAQs are explicitly about things we did at school. Why don’t we add fractions the way we multiply them? What is point nine recurring? People often ask these questions, and this seemed a good place to explain the thinking behind them. Which is not always what you might expect, and in one case not what I expected when I started to write that item, thanks to a coincidental email that changed my mind.

  However, school mathematics is only a tiny part of a much greater enterprise, which spans millennia of human culture and ranges over the entire planet. Mathematics is essential to virtually everything that affects our lives - mobile phones, medicine, climate change - and it is growing faster than it has ever done before. But most of this activity goes on behind the scenes, and it’s all too easy to assume that it’s not happening at all. So, in Hoard I’ve devoted a bit more space to quirky or unusual applications of mathematics, both in everyday life and in frontier science. And a bit less to the big problems of pure mathematics, mainly because I covered several of the really juicy ones in Cabinet.

  These items range from finding the area of an ostrich egg to the puzzling excess of matter over antimatter just after the Big Bang. And I’ve also included a few historical topics, like Babylonian numerals, the abacus and Egyptian fractions. The history of mathematics goes back at least 5,000 years, and discoveries made in the distant past are still important today, because mathematics builds on its past successes.

  A few items are longer than the rest - mini-essays about important topics that you may have come across in the news, like the fourth dimension or symmetry or turning a sphere inside out. These items don’t exactly go beyond school mathematics: they generally head off in a completely different direction. There is far more to mathematics than most of us realise. I’ve also deposited a few technical comments in the notes, which are scattered among the answers. These are things I felt needed to be s
aid, and needed to be easy to ignore. I’ve given cross-references to Cabinet where appropriate.

  Occasionally you may come across a complicated-looking formula - though most of those have been relegated to the notes at the back of the book. If you hate formulas, skip these bits. The formulas are there to show you what they look like, not because you’re going to have to pass a test. Some of us like formulas - they can be extraordinarily pretty, though they are admittedly an acquired taste. I didn’t want to cop out by omitting crucial details; I personally find this very annoying, like the TV programmes that bang on about how exciting some new discovery is, but don’t actually tell you anything about it.

  Despite its random arrangement, the best way to read Hoard is probably to do the obvious: start at the front and work your way towards the back. That way you won’t end up reading the same page six times while missing out on something far more interesting. But you should feel positively eager to skip to the next item the moment you feel you’ve wandered into the wrong drawer by mistake.

  This is not the only possible approach. For much of my professional life, I have read mathematics books by starting at the back, thumbing towards the front until I spot something that looks interesting, continuing towards the front until I find the technical terms upon which that thing depends, and then proceeding in the normal front-to-back direction to find out what’s really going on.

  Well, it works for me. You may prefer a more conventional approach.

  Ian Stewart

  Coventry, April 2009

  A mathematician is a machine for turning coffee into theorems.

  Paul Erdős

  To Avril, for 40 years of devotion and support

  Calculator Curiosity 1

  Get your calculator, and work out:

  (8×8) + 13

  (8×88) + 13

  (8×888) + 13

  (8×8888) + 13

  (8×88888) + 13

  (8×888888) + 13

  (8×8888888) + 13

  (8×88888888) + 13

  Answers on page 274

  Year Turned Upside Down

  Some digits look (near enough) the same upside down: 0, 1, 8. Two more come in a pair, each the other one turned upside down: 6, 9. The rest, 2, 3, 4, 5, 7, don’t look like digits when you turn them upside down. (Well, you can write 7 with a squiggle and then it looks like 2 upside down, but please don’t.) The year 1691 reads the same when you turn it upside down.

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