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 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

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

Start Here

•••  There are three kinds of people

in the world:

those who can count,

and those who can’t.

When I was fourteen years old, I started a notebook. A math notebook. Before you write me off as a sad case, I hasten to add that it wasn’t a notebook of school math. It was a notebook of every interesting thing I could find out about the math that wasn’t 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 math you did at school is not all of it. Better still: the math you didn’t do at school is interesting. In fact, a lot of it is fun—especially when you don’t 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 favorites, which tend to reappear from time to time and often cause renewed excitement when they do—the car and the goats, and the 12-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 is new, specially designed for this book. I’ve striven for variety, so there are logic puzzles, geometric 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 math is that you can impress the hell out of your friends. (Be modest about it, though, that’s 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 on the latest buzzwords. So I’ve scattered some short “essays” here and there, written in an informal, nontechnical style. The essays explain some of the recent breakthroughs that have featured prominently in the media. Things like Fermat’s Last Theorem—remember the TV program? And the Four-Color Theorem, the Poincaré Conjecture, Chaos Theory, Fractals, Complexity Science, Penrose Patterns. Oh, and there are also some unsolved questions, just to show that math isn’t 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: π, prime numbers, Pythagoras’ 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, it’s 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. I’ve made sure that a lot of easy stuff is mixed in everywhere, though, to avoid wearing your brain out too quickly.

What I’m 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 I’d 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

Alien Encounter

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, they’ll 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.’

‘They are all hermandrofemigynes,’ said Crock.

‘Whatever. Now, Alfy: to which species does Betty belong?’

‘Gibberish.’

‘Ah. Betty: do Alfy and Gemma belong to different species?’

‘No.’

‘Right . . . Talkative lot, aren’t they? Um . . . Gemma: to which species does Betty belong?’

‘Veracitor.’

Quirk nodded knowledgeably. ‘Right, that’s settled it, then!’

‘Settled what, Captain?’

‘Which species each belongs to.’

‘I see. And those species are—?’

‘Haven’t the foggiest idea, Crock. You’re the one who’s supposed to be logical!’

Answer on page 252

Tap-an-Animal

This is a great mathematical party trick for children. They take turns to choose an animal. Then they spell out its name while you, or another child, tap successive points of the ten-pointed star. You must start at the point labelled ‘Rhinoceros’, and move in a clockwise direction along the lines. Miraculously, as they say the final letter, you tap the correct animal.

Spell the name to find the animal.

How does it work? Well, the third word along the star is ‘Cat’, which has three letters, the fourth is ‘Lion’, with four, and so on. To help conceal the trick, the animals in positions 0, 1 and 2 have 10, 11 and 12 letters. Since 10 taps takes you back to where you started, everything works out perfectly.

To conceal the trick, use pictures of the animals – in the diagram I’ve used their names for clarity.

Curious Calculations

Your calculator can do tricks.

(1) Try these multiplications. What do you notice?

Does the pattern continue if you use longer strings of 1’s?

(2) Enter the number

           142,857

(preferably into the memory) and multiply it by 2, 3, 4, 5, 6 and 7. What do you notice?

Answers on page 252

Triangle of Cards

I have 15 cards, numbered consecutively from 1 to 15. I want to lay them out in a triangle. I’ve put numbers on the top three for later reference:

Triangle of cards.

However, I don’t want any old arrangement. I want each card to be the difference between the two cards immediately below it, to left and right. For example, 5 is the difference between 4 and 9. (The differences are always calculated so that they are positive.) This condition does not apply to the cards in the bottom row, you appreciate.

The top three cards are already in place – and correct. Can you find how to place the remaining twelve cards?

Mathematicians have found ‘difference triangles’ like this with two, three or four rows of cards, bearing consecutive whole numbers starting from 1. It has been proved that no difference triangle can have six or more rows.

Answer on page 253

Pop-up Dodecahedron

The dodecahedron is a solid made from twelve pentagons, and is one of the five regular solids (page 174).

Three stages in making a pop-up dodecahedron.

Cut out two identical copies of the left-hand diagram – 10cm across – from thickish card. Crease heavily along the joins so that the five pentagonal flaps are nice and floppy. Place one copy on top of the other, like the centre diagram. Lace an elastic band alternately over and under, as in the right-hand diagram (thick solid lines show where the band is on top) – while holding the pieces down with your finger.

Now let go.

If you’ve got the right size and strength of elastic band, the whole thing will pop up to form a three-dimensional dodecahedron.

Popped-up dodecahedron.

Sliced Fingers

Here’s how to wrap a loop of string around somebody’s fingers – your own or those of a ‘volunteer’ – so that when it is pulled tight it seems to slice through the fingers. The trick is striking because we know from experience that if the string is genuinely linked with the fingers then it shouldn’t slip off. More precisely, imagine that your fingers all touch a fixed surface – thereby preventing the string from sliding off their tips. The trick is equivalent to removing the loop from the holes created by your fingers and the surface. If the loop were really linked through those holes, you couldn’t remove it at all, so it has to appear to be linked without actually being linked.

If by mistake it is linked, it really would have to slice through your fingers, so be careful.

How (not) to slice your fingers off.

Why is this a mathematical trick? The connection is topology, a branch of mathematics that emerged over the past 150 years, and is now central to the subject. Topology is about properties such as being knotted or linked – geometrical features that survive fairly drastic transformations. Knots remain knotted even if the string is bent or stretched, for instance.

Make a loop from a 1-metre length of string. Hook one end over the little finger of the left hand, twist, loop it over the next finger, twist in the same direction, and keep going until it passes behind the thumb (left picture). Now bring it round in front of the thumb, and twist it over the fingers in reverse order (right picture). Make sure that when coming back, all the twists are in the opposite direction to what they were the first time.

Fold the thumb down to the palm of the hand, releasing the string. Pull hard on the free loop hanging from the little finger . . . and you can hear it slice through those fingers. Yet, miraculously, no damage is done.

Unless you get a twist in the wrong direction somewhere.

Turnip for the Books

‘It’s been a good year for turnips,’ farmer Hogswill remarked to his neighbour, Farmer Suticle.

‘Yup, that it has,’ the other replied. ‘How many did you grow?’

‘Well . . . I don’t exactly recall, but I do remember that when I took the turnips to market, I sold six-sevenths of them, plus one-seventh of a turnip, in the first hour.’

‘Must’ve been tricky cuttin’ ’em up.’

‘No, it was a whole number that I sold. I never cuts ’em.’

‘If’n you say so, Hogswill. Then what?’

‘I sold six-sevenths of what was left, plus one-seventh of a turnip, in the second hour. Then I sold six-sevenths of what was left, plus one-seventh of a turnip, in the third hour. And finally I sold six-sevenths of what was left, plus one-seventh of a turnip, in the fourth hour. Then I went home.’

‘Why?’

“Cos I’d sold the lot.’

How many turnips did Hogswill take to market?

Answer on page 253

The Four-Colour Theorem

Problems that are easy to state can sometimes be very hard to answer. The four-colour theorem is a notorious example. It all began in 1852 with Francis Guthrie, a graduate student at University College, London. Guthrie wrote a letter to his younger brother Frederick, containing what he thought would be a simple little puzzle. He had been trying to colour a map of the English counties, and had discovered that he could do it using four colours, so that no two adjacent counties were the same colour. He wondered whether this fact was special to the map of England, or more general. ‘Can every map drawn on the plane be coloured with four (or fewer) colours so that no two regions having a common border have the same colour?’ he wrote.

It took 124 years to answer him, and even now, the answer relies on extensive computer assistance. No simple conceptual proof of the four-colour theorem – one that can be checked step by step by a human being in less than a lifetime – is known.

Colouring England’s counties with four colours – one solution out of many.

Frederick Guthrie couldn’t answer his brother’s question, but he ‘knew a man who could’ – the famous mathematician Augustus De Morgan. However, it quickly transpired that De Morgan couldn’t, as he confessed in October of the same year in a letter to his even more famous Irish colleague, Sir William Rowan Hamilton.

It is easy to prove that at least four colours are necessary for some maps, because there are maps with four regions, each adjacent to all the others. Four counties in the map of England (shown here slightly simplified) form such an arrangement, which proves that at least four colours are necessary in this case. Can you find them on the map?

A simple map needing four colours.

De Morgan did make some progress: he proved that it is not possible to find an analogous map with five regions, each adjacent to all four of the others. However, this does not prove the four-colour theorem. All it does is prove that the simplest way in which it might go wrong doesn’t happen. For all we know, there might be a very complicated map with, say, a hundred regions, which can’t be coloured using only four colours because of the way long chains of regions connect to their neighbours. There’s no reason to suppose that a ‘bad’ map has only five regions.

The first printed reference to the problem dates from 1878, when Arthur Cayley wrote a letter to the Proceedings of the London Mathematical Society (a society founded by De Morgan) to ask whether anyone had solved the problem yet. They had not, but in the following year Arthur Kempe, a barrister, published a proof, and that seemed to be that.

Kempe’s proof was clever. First he proved that any map contains at least one region with five or fewer neighbours. If a region has three neighbours, you can shrink it away, getting a simpler map, and if the simpler map can be 4-coloured, so can the original one. You just give the region that you shrunk whichever colour differs from those of its three neighbours. Kempe had a more elaborate method for getting rid of a region with four or five neighbours. Having established this key fact, the rest of the proof was straightforward: to 4-colour a map, keep shrinking it, region by region, until it has four regions or fewer. Colour those regions with different colours, and then reverse the procedure, restoring regions one by one and colouring them according to Kempe’s rules. Easy!

If the right-hand map can be 4-coloured, so can the left-hand one.

It looked too good to be true – and it was. In 1890 Percy Heawood discovered that Kempe’s rules didn’t always work. If you shrunk a region with five neighbours, and then tried to put it back, you could run into terminal trouble. In 1891 Peter Guthrie Tait thought he had fixed this error, but Julius Petersen found a mistake in Tait’s method, too.

Heawood did observe that Kempe’s method can be adapted to prove that five colours are always sufficient for any map. But no one could find a map that needed more than four. The gap was tantalising, and quickly became a disgrace. When you know that a mathematical problem has either 4 or 5 as its answer, surely you ought to be able to decide which!

But . . . no one could.

The usual kind of partial progress then took place. In 1922 Philip Franklin proved that all maps with 26 or fewer regions can be 4-coloured. This wasn’t terribly edifying in itself, but Franklin’s method paved the way for the eventual solution by introducing the idea of a reducible configuration. A configuration is any connected set of regions within the map, plus some information about how many regions are adjacent to those in the configuration. Given some configuration, you can remove it from the map to get a simpler map – one with fewer regions. The configuration is reducible if there is a way to 4-colour the original map, provided you can 4-colour the simpler map. In effect, there has to be a way to ‘fill in’ colours in that configuration, once everything else has been 4-coloured.

A single region with only three neighbours forms a reducible configuration, for instance. Remove it, and 4-colour what’s left – if you can. Then put that region back, and give it a colour that has not been used for its three neighbours. Kempe’s failed proof does establish that a region with four neighbours forms a reducible configuration. Where he went wrong was to claim the same thing for a region with five neighbours.

Franklin discovered that configurations containing several regions can sometimes work when single regions don’t. Lots of multi-region configurations turn out to be reducible.

Kempe’s proof would have worked if every region with five neighbours were reducible, and the reason why it would have worked is instructive. Basically, Kempe thought he had proved two things. First, every map contains a region with either three, four or five adjacent ones. Second, each of the associated configurations is reducible. Now these two facts together imply that every map contains a reducible configuration. In particular, when you remove a reducible configuration, the resulting simpler map also contains a reducible configuration. Remove that one, and the same thing happens. So, step by step, you can get rid of reducible configurations until the result is so simple that it has at most four regions. Colour those however you wish – at most four colours will be needed. Then restore the previously removed configuration; since this was reducible, the resulting map can also be 4-coloured . . . and so on. Working backwards, we eventually 4-colour the original map.

This argument works because every map contains one of our irreducible configurations: they form an ‘unavoidable set’.

Kempe’s attempted proof failed because one of his configurations, a region with five neighbours, isn’t reducible. But the message from Franklin’s investigation is: don’t worry. Try a bigger list, using lots of more complicated configurations. Dump the region with five neighbours; replace it by several configurations with two or three regions. Make the list as big as you need. If you can find some unavoidable set of reducible configurations, however big and messy, you’re done.

In fact – and this matters in the final proof – you can get away with a weaker notion of unavoidability, applying only to ‘minimal criminals’: hypothetical maps that require five colours, with the nice feature that any smaller map needs only four colours. This condition makes it easier to prove that a given set is unavoidable. Ironically, once you prove the theorem, it turns out that no minimal criminals exist. No matter: that’s the proof strategy.

In 1950 Heinrich Heesch, who had invented a clever method for proving that many configurations are reducible, said that he believed the four-colour theorem could be proved by finding an unavoidable set of reducible configurations. The only difficulty was to find one – and it wouldn’t be easy, because some rule-of-thumb calculations suggested that such a set would have to include about 10,000 configurations.

By 1970 Wolfgang Haken had found some improvements to Heesch’s method for proving configurations to be reducible, and began to feel that a computer-assisted proof was within reach. It should be possible to write a computer program to check that each configuration in some proposed set is reducible. You could write down several thousand configurations by hand, if you really had to. Proving them unavoidable would be time-consuming, but not necessarily out of reach. But with the computers then available, it would have taken about a century to deal with an unavoidable set of 10,000 configurations. Modern computers can do the job in a few hours, but Haken had to work with what was available, which meant that he had to improve the theoretical methods, and cut the calculation down to a feasible size.