Professor Stewart's Hoard of Mathematical Treasures (2 page)

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
said, 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

Get your calculator, and work out:

Answers on page 274

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.

Which year in the past is the most recent one that reads the same when you turn it upside down?

Which year in the future is the next one that reads the same when you turn it upside down?

 

Answers on page 274

Lilavati.

Among the great mathematicians of ancient India was Bhaskara, ‘The Teacher’, who was born in 1114. He was really an astronomer: in his culture, mathematics was mainly an astronomical technique. Mathematics appeared in astronomy texts; it was not a separate subject. Among Bhaskara’s most famous works is one named Lilavati. And thereby hangs a tale.

Fyzi, Court Poet to the Mogul Emperor Akbar, wrote that Lilavati was Bhaskara’s daughter. She was of marriageable age, so Bhaskara cast her horoscope to work out the most propitious wedding date. (Right into the Renaissance period, many mathematicians made a good living casting horoscopes.) Bhaskara, clearly a bit of a showman, thought he’d come up with a terrific idea to make his forecast more dramatic. He bored a hole in a cup and floated it in a bowl of water, with everything designed so that the cup would sink when the fateful moment arrived.

Unfortunately, an eager Lilavati was leaning over the bowl, hoping that the cup would sink. A pearl from her dress fell into the cup and blocked the hole. So the cup didn’t sink, and poor Lilavati could never get married.

To cheer his daughter up, Bhaskara wrote a mathematics textbook for her.

Hey, thanks, Dad.

Sixteen matches are arranged to form five identical squares.

By moving exactly two matches, reduce the number of squares to four. All matches must be used, and every match should be part of one of the squares.

 

Answer on page 274

Sixteen matches arranged to form five squares.

Elephants always wear pink trousers.
Every creature that eats honey can play the bagpipes.
Anything that is easy to swallow eats honey.

No creature that wears pink trousers can play the bagpipes.

Therefore:

Elephants are easy to swallow.

Is the deduction correct, or not?

 

Answer on page 274

In the diagram there are three big circles, and each passes through four small circles. Place the numbers 1, 2, 3, 4, 5, 6 in the small circles so that the numbers on each big circle add up to 14.

 

Answer on page 276

Make the sum 14 around each big circle.

This is a mathematical game with very simple rules that’s a lot of fun to play, even on a small board. It was invented by puzzle expert and writer Colin Vout. The picture shows the 4×4 case.