Ian Stewart - Professor Stewart’s Cabinet of Mathematical Curiosities

By the Same Author

Concepts of Modern Mathematics

Game, Set, and Math

Does God Play Dice?

Another Fine MathYouveGot Me Into

Fearful Symmetry (with Martin Golubitsky)

NaturesNumbers

From Here to Infinity

The Magical Maze

LifesOther Secret

Flatterland

What Shape is a Snowflake?

The Annotated Flatland (with Edwin A. Abbott)

Math Hysteria

The Mayor ofUglyvillesDilemma

Letters to a Young Mathematician

How to Cut a Cake

Why Beauty is Truth

Taming the Infinite

Professor Stewarts
Cabinet of
Mathematical Curiosities

Ian Stewart

First published in Great Britain in 2008 by
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Contents

When I was fourteen years old, I started a notebook. A maths notebook. Before you write me off as a sad case, I hasten to add that it wasnt a notebook of school maths. It was a notebook of every interesting thing I could find about the maths that wasnt taught at school. Which, I discovered, was a lot, because I soon had to buy another notebook.

OK, now you can write me off. But before you do, have you spotted the messages in this sad little tale? The maths you did atschool is not all of it. Better still: the maths you didnt do at school isinteresting. In fact, a lot of it is fun especially when you dont have to pass a test or get the sums right.

My notebook grew to a set of six, which I still have and then spilled over into a filing cabinet when I discovered the virtues of the photocopier. Curiosities is a sample from my cabinet, a miscellany of intriguing mathematical games, puzzles, stories and factoids. Most items stand by themselves, so you can dip in at almost any point. A few form short mini-series. I incline to the view that a miscellany should be miscellaneous, and this one is.

The games and puzzles include some old favourites, which tend to reappear from time to time, and often cause renewed excitement when they do the car and the goats, and the twelve ball weighing puzzle, both caused a huge stir in the media: one in the USA, the other in the UK. A lot of the material here is new, specially designed for this book. Ive striven for variety, so there are logic puzzles, geometrical puzzles, numerical puzzles, probability puzzles, odd items of mathematical culture, things to do and things to make.

One of the virtues of knowing a bit of maths is that you can impress the hell out of your friends. (Be modest about it, though, thats my advice. You can also annoy the hell out of your friends.) A good way to achieve this desirable goal is to be up to speed with the latest buzzwords. So Ive scattered some short essays here and there, written in an informal, non-technical style. The essays explain some of the recent breakthroughs that have featured prominently in the media. Things like Fermats Last Theorem remember the TV programme? And the four-colour theorem, the Poincar Conjecture, chaos theory, fractals, complexity science, Penrose patterns. Oh, and there are also some unsolved questions, just to show that maths isnt all done. Some are recreational, some serious like the P = NP? problem, for which a million-dollar prize is on offer. You may not have heard of the problem, but you need to know about the prize.

Shorter, snappy sections reveal interesting facts and discoveries about familiar but fascinating topics: , prime numbers, Pythagorass Theorem, permutations, tilings. Amusing anecdotes about famous mathematicians add a historical dimension and give us all a chance to chuckle sympathetically at their endearing foibles.

Now, I did say you could dip in anywhere and you can, believe me but to be brutally honest, its probably better to start at the beginning and dip in following much the same order as the pages. A few of the early items help with later ones, you see. And the early ones tend to be a bit easier, while some of the later ones are, well, a bit... challenging. Ive made sure that a lot of easy stuff is mixed in everywhere, though, so that you dont wear your brain out too quickly.

What Im trying to do is to excite your imagination by showing you lots of amusing and intriguing pieces of mathematics. I want you to have fun, but Id also be overjoyed if Curiosities encouraged you to engage with mathematics, experience the thrill of discovery, and keep yourself informed about important developments be they from four thousand years ago, last week or tomorrow.

Ian Stewart

Coventry, January 2008

The starship Indefensible was in orbit around the planet Noncomposmentis, and Captain Quirk and Mr Crock had beamed down to the surface.

According to the Good Galaxy Guide, there are two species of intelligent aliens on this planet, said Quirk.

Correct, Captain Veracitors and Gibberish. They all speak Galaxic, and they can be distinguished by how they answer questions. The Veracitors always reply truthfully, and the Gibberish always lie.

But physically

they are indistinguishable, Captain.

Quirk heard a sound, and turned to find three aliens creeping up on them. They looked identical.

Welcome to Noncomposmentis, said one of the aliens.

I thank you. My name is Quirk. Now, you are... Quirk paused. No point in asking their names, he muttered. For all we know, theyll be wrong.

That is logical, Captain, said Crock.

Because we are poor speakers of Galaxic, Quirk improvised, I hope you will not mind if I call you Alfy, Betty and Gemma. As he spoke, he pointed to each of them in turn. Then he turned to Crock and whispered, Not that we know what sex they are, either.