Ian stewart, p.

**1**## Ian Stewart, page 1

Table of Contents

By the Same Author

Title Page

Acknowledgements

Second Drawer Down . . .

Calculator Curiosity 1

Year Turned Upside Down

Luckless Lovelorn Lilavati

Sixteen Matches

Swallowing Elephants

Magic Circle

Dodgem

Press-the-Digit-ation

Secrets of the Abacus

Redbeard’s Treasure

Hexaflexagons

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

Flatterland

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

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

Acknowledgements

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

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.