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
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
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
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.
Ian Stewart Page 1
