Professor Stewart's Hoard of Mathematical Treasures

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 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

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.

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

Luckless Lovelorn Lilavati

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

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.

Swallowing Elephants

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

Magic Circle

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.

Dodgem

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.

Dodgem on a 4 × 4 board.

Players take turns to move one of their counters one cell forward, one to the left, or one to the right, as shown by the arrows ‘black’s directions’ and ‘white’s directions’. They can’t move a counter if it is blocked by an opponent’s counter or the edge of the board, except for the opposite edge where their counters can escape. A player must always leave his opponent at least one legal move, and loses the game if he does not. The first player to make all his counters escape wins.

On a larger board the initial arrangement is similar, with the lower left-hand corner unoccupied, a row of white counters up the left-hand column, and a row of black counters along the bottom row.

Vout proved that with perfect strategy the first player always wins on a 3 × 3 board, but for larger boards it seems not to be known who should win. A good way to play is with draughts (checkers) pieces on the usual 8 × 8 board.

It seems natural to use square boards, but with a rectangular board the player with fewer counters has to move them further, so the game may be playable on rectangular boards. As far as I know, they’ve not been considered.

Press-the-Digit-ation

I learned this trick from the Great Whodunni, a hitherto obscure stage magician who deserves wider recognition. It’s great for parties, and only the mathematicians present will guess how it works.* It is designed specifically to be used in the year 2009, but I’ll explain how to change it for 2010, and the Answer section on page 276 will extend that to any year.

Whodunni asks for a volunteer from the audience, and his beautiful assistant Grumpelina hands them a calculator. Whodunni then makes a big fuss about it having once been a perfectly ordinary calculator, until it was touched by magic. Now, it can reveal your hidden secrets.

‘I am going to ask you to do some sums,’ he tells them. ‘My magic calculator will use the results to display your age and the number of your house.’ Then he tells them to perform the following calculations:

•    Enter your house number.

•    Double it.

•    Add 42.

•    Multiply by 50.

•    Subtract the year of your birth.

•    Subtract 50.

•    Add the number of birthdays you have had so far this year, i.e. 0 or 1.

•    Subtract 42.

‘I now predict,’ says Whodunni, ‘that the last two digits of the result will be your age, and the remaining digits will be the number of your house.’

Let’s try it out on the fair Grumpelina, who lives in house number 327. She was born on 31 December 1979, and let’s suppose that Whodunni performs his trick on Christmas Day 2009, when she is 29.

•    Enter your house number: 327

•    Double it: 654.

•    Add 42: 696.

•    Multiply by 50: 34,800.

•    Subtract the year of your birth: 32,821

•    Subtract 50: 32,771.

•    Add the number of birthdays you have had so far this year (0): 32,771.

•    Subtract 42: 32,729.

The last two digits are 29, Grumpelina’s age. The others are 327 – her house number.

The trick works for anyone aged 1 to 99, and any house number, however large. You could ask for a phone number instead, and it would still work. But Grumpelina’s phone number is ex-directory, so I can’t illustrate the trick with that.

If you’re trying the trick in 2010, replace the last step by ‘subtract 41’.

You don’t need a magic calculator, of course: an ordinary one works fine. And you don’t need to understand how the trick works to amaze your friends. But for those who’d like to know the secret, I’ve explained it on page 276.

Secrets of the Abacus

In these days of electronic calculators, the device known as an abacus seems rather outmoded. Most of us encounter it as a child’s educational toy, an array of wires with beads that slide up and down to represent numbers. However, there’s more to the abacus than that, and such devices are still widely used, mainly in Asia and Africa. For a history, see: en.wikipedia.org/wiki/Abacus

The basic principle of the abacus is that the number of beads on each wire represents one digit in a sum, and the basic operations of arithmetic can be performed by moving the beads in the right way. A skilled operator can add numbers together as fast as someone using a calculator can type them in, and more complicated things like multiplication are entirely practical.

The Sumerians used a form of abacus around 2500 BC, and the Babylonians probably did too. There is some evidence of the abacus in ancient Egypt, but no images of one have yet been found, only discs that might have been used as counters. The abacus was widely used in the Persian, Greek and Roman civilisations. For a long time, the most efficient design was the one used by the Chinese from the 14th century onwards, called a suànpán. It has two rows of beads; those in the lower row signify 1 and those in the upper row signify 5. The beads nearest the dividing line determine the number. The suànpán was quite big: about 20 cm (8 inches) high and of varying width depending on the number of columns. It was used lying flat on a table to stop the beads sliding into unwanted positions.

The number 654,321 on a Chinese abacus.

The Japanese imported the Chinese abacus from 1600, improved it to make it smaller and easier to use, and called it the soroban. The main differences were that the beads were hexagonal in cross-section, everything was just the right size for fingers to fit, and the abacus was used lying flat. Around 1850 the number of beads in the top row was reduced to one, and around 1930 the number in the bottom row was reduced to four.

Japanese abacus, cleared.

The first step in any calculation is to clear the abacus, so that it represents 0 … 0. To do this efficiently, tilt the top edge up so that all the beads slide down. Then lie the abacus flat on the table, and run your finger quickly along from left to right, just above the dividing line, pushing all the top beads up.

Japanese abacus, representing 9,876,543,210.

Again, numbers in the lower row signify 1 and those in the upper row signify 5. The Japanese designer made the abacus more efficient by removing superfluous beads that provided no new information.

The operator uses the soroban by resting the tips of the thumb and index finger lightly on the beads, one either side of the central bar, with the hand hovering over the bottom rows of beads. Various ‘moves’ must then be learned, and practised, much like a musician learns to play an instrument. These moves are the basic components of an arithmetical calculation, and the calculation itself is rather like playing a short ‘tune’. You can find lots of detailed abacus techniques at: www.webhome.idirect.com/~totton/abacus/pages.htm#Soroban1

I’ll mention only the two easiest ones.

A basic rule is: always work from left to right. This is the opposite of what we teach in school arithmetic, where the calculation proceeds from the units to the tens, the hundreds, and so on – right to left. But we say the digits in the left–right order: ‘three hundred and twenty-one’. It makes good sense to think of them that way, and to calculate that way. The beads act as a memory, too, so that you don’t get confused by where to put the ‘carry digits’.

To add 572 and 142, for instance, follow the instructions in the pictures. (I’ve referred to the columns as 1, 2, 3, from the right, because that’s the way we think. The fourth column doesn’t play any role, but it would do if we were adding, say, 572 and 842, where 5 + 8 = 13 involves a ‘carry digit’ 1 in place 4.

A basic technique occurs in subtraction. I won’t draw where the beads go, but the principle is this. To subtract 142 from 572, change each digit x in 142 to its complement 10 − x. So 142 becomes 968. Now add 968 to 572, as before. The result is 1,540, but of course 572 − 142 is actually 430. Ah, but I haven’t yet mentioned that at each step you subtract 1 from the column one place to the left (doing this as you proceed). So the initial 1 disappears, 5 turns into 4, and 4 turns into 3. The 0 you leave alone.

Why does this work, and why do we leave the units digit unchanged?

Answer on page 277

Redbeard’s Treasure

Captain ‘Jolly’ Roger Redbeard, the fiercest pirate in the Windlass Islands, stared blankly at a diagram he had drawn in the sand beside the quiet lagoon behind Rope’s End Reef. He had buried a hoard of pieces of eight there a few years ago, and now he wanted to retrieve his treasure. But he had forgotten where it was. Fortunately he had set up a cunning mnemonic, to remind him. Unfortunately, it was a bit too cunning.

He addressed the band of tattered thugs that constituted his crew.

‘Avast, ye stinkin’ bilge-rats! Oi, Numbskull, put down that cask o’ rotgut and listen!’

The crew eventually quietened down.

‘You remember when we boarded the Spanish Prince? And just before I fed the prisoners to the sharks, one of ’em told us where they’d hidden their loot? An’ we dug it all up and reburied it somewhere safe?’

There was a ragged cry, mostly of agreement.

‘Well, the treasure is buried due north o’ that skull-shaped rock over there. All we need to know is how far north. Now, I ’appens to know that the exact number o’ paces is the number of different ways a man can spell out the word TREASURE by puttin’ his finger on the T at the top o’ this diagram, and then movin’ it down one row at a time to a letter that’s next to it, one step to the left or right.

‘I’m offerin’ ten gold doubloons to the first man-jack o’ ye to tell me that number. What say ye, lads?’

T

R  R

E  E  E

A  A  A  A

S  S  S  S  S

U  U  U  U  U  U

R  R  R  R  R  R  R

E  E  E  E  E  E  E  E

How many paces is it from the rock to the treasure?

Answer on page 277

Hexaflexagons