So You Think You've Got Problems Read online

  To Natalie

  CONTENTS

  TITLE PAGE

  DEDICATION

  INTRODUCTION

  Tasty teasers: number conundrums

  The puzzle zoo

  ANIMAL PROBLEMS

  Tasty teasers: gruelling grids

  I’m a mathematician, get me out of here

  SURVIVAL PROBLEMS

  Tasty teasers: riotous riddles

  Cakes, cubes and a cobbler’s knife

  GEOMETRY PROBLEMS

  Tasty teasers: pencils and utensils

  A wry plod

  WORD PROBLEMS

  Tasty teasers: Bongard bafflers

  Sleepless nights and sibling rivalries

  PROBABILITY PROBLEMS

  ANSWERS

  Tasty Teasers: Number conundrums

  The puzzle zoo

  ANIMAL PROBLEMS

  Tasty teasers: Gruelling grids

  I’m a mathematician, get me out of here

  SURVIVAL PROBLEMS

  Tasty teasers: Riotous riddles

  Cakes, cubes and a cobbler’s knife

  GEOMETRY PROBLEMS

  Tasty teasers: Pencils and utensils

  A wry plod

  WORD PROBLEMS

  Tasty teasers: Bongard bafflers

  Sleepless nights and sibling rivalries

  PROBABILITY PROBLEMS

  A LIST OF THE PUZZLES AND A NOTE ON THEIR SOURCES

  ACKNOWLEDGEMENTS

  ABOUT THE AUTHOR

  COPYRIGHT

  INTRODUCTION

  Archimedes was the greatest scientist of antiquity. He made stunning theoretical discoveries about concepts such as pi and infinity, and designed some of the most advanced technology of his age.

  He is also responsible for one of the worst puzzles in history.

  It’s a stinker, believe me.

  The ‘cattle problem’ is difficult, inelegant and absurd. It is nevertheless a perfect place to begin a book of recreational problems. For a start: Archimedes. Secondly, it is a fascinating historical oddity. Lastly, bad puzzles illuminate what makes good ones good. Our dissection of the Archimedean herd explains what the rest of the puzzles in this book will not be like. You will be grateful.

  THE CATTLE PROBLEM

  The sun god had a herd of cattle that grazed on the plains of Sicily. The bulls and cows came in four colours: white, black, yellow and dappled, such that the numbers of each type could be expressed in the following way.

  White bulls = (1/2+1/3) black bulls + yellow bulls,

  Black bulls = (1/4+1/5) dappled bulls + yellow bulls,

  Dappled bulls = (1/6+1/7) white bulls + yellow bulls,

  White cows = (1/3+1/4) black herd,

  Black cows = (1/4+1/5) dappled herd,

  Dappled cows = (1/5+1/6) yellow herd,

  Yellow cows = (1/6+1/7) white herd,

  White bulls + black bulls = a square number,

  Dappled bulls + yellow bulls = a triangular number.

  What is the total size of the herd?

  Urgh. Before we try to digest this unappetising soup of fractions, here’s its curious backstory. The problem was discovered in a library in Germany in the eighteenth century, two thousand years after Archimedes died. Written in Greek in the form of a poem made up of 22 couplets, it was found in a manuscript that no one had looked at before, accompanied by a note stating that Archimedes had sent it to Eratosthenes, head of the library at Alexandria.

  On the positive side, the problem is set in verse. At least its author aimed to entertain. The maths, on the other hand, is less jolly. The cattle problem demands a farmyard of algebra. The first seven lines can be written as seven equations in eight unknowns. With enough patience and enough paper, after much tedious computation and shuffling around of variables, you will discover that the smallest possible size of the herd that satisfies the first seven lines is 50,389,082. (Which means Sicily would have a bull or a cow for every 500 square metres.)

  If you’ve solved the puzzle thus far, Archimedes congratulates you. But don’t get smug. ‘Thou … can not be regarded as of high skill,’ he warns. We’ve not got to the hard bit yet.

  The eighth line states that the number of white and black bulls is a square number, meaning a number like 1, 4, 9 or 16 that is the square of another number (i.e., 12, 22, 32, 42). If this property is included, the smallest possible size of the herd is 51,285,802,909,803. (Sicily now has about 2,000 animals per square metre, meaning that the island is entirely covered with cattle, squashed like sardines and stacked hundreds of metres high.) This calculation requires a little bit more advanced algebra, but not too much. Gotthold Ephraim Lessing, the German librarian who discovered the problem, showed it to a mathematician friend who was able to come up with this solution.

  The final line is the killer. It states that the number of dappled and yellow bulls is a triangular number, meaning a number that can be arranged in a dot triangle, such as 3, 6, 10 – as in – and so on, with an extra line each time. Bam! Archimedes’ cattle problem is now beyond the scope of eighteenth-century mathematics.

  For the next hundred years, the bovine brainteaser was a celebrated unsolved problem. It was rumoured that Carl Friedrich Gauss, the greatest mathematician of the nineteenth century, had solved it completely. The first person to publish a partial solution, however, was fellow German August Amthor in 1880, who revealed that the smallest possible herd was a number beginning with 766 and continuing for another 206,542 digits. In other words, a number so ridiculously huge that the universe would not be able to contain this herd even if every bull and cow was as small as an atom.

  Undeterred by the scale of the job, in 1889 three friends in Illinois with nothing better to do started to work out the other digits. After four years’ toil they had calculated 32 of the digits on the left side of the number and 12 on the right. The full solution to the cattle problem, however, required the arrival of the computer age. In 1965 a supercomputer took 7 hours and 45 minutes to print out the number, which ran to 42 sheets of A4.

  Lessing and others have questioned whether Archimedes was indeed the author of the cattle problem. No reference to the puzzle exists in any other Greek writing, and Archimedes could not possibly have known the answer to the question he supposedly set. Yet some academics are convinced it does date to him. Archimedes was fascinated by extraordinarily large numbers; in his short text, The Sand Reckoner, he devises a new number system in order to estimate the number of grains of sand that would fill the universe. (His estimate: 1063 grains). Perhaps the point of the cattle problem was not to solve it at all, but to show how nine simple statements using unit fractions could determine a number that (in Archimedes’s time) was beyond all comprehension. To concoct a whimsical, easily understandable problem that nevertheless remains unsolved for more than 2,000 years is arguably the mark of (evil) genius. ‘If thou hast computed [the answer], O friend, and found the total number of cattle,’ ends the verse, ‘then exult as a conqueror, for thou hast proved thyself most skilled in numbers.’ Quite.

  As a puzzle, the cattle problem is less a piece of recreational maths than an overly complicated exercise in solving simultaneous equations.

  The remaining puzzles in this book:

  Prize insight over computation.

  Engage basic competence rather than technical skills.

  Deal in numbers that can be written down on fewer than 42 sheets of A4.

  Can be fully solved in less than 2,000 years.

  I’ll take my lead from Archimedes in only one way: his zoological choice of subject matter.

  This book kicks off with a chapter of puzzles about animals. Randy rabbits, mischievous moggies, frogs, flies, lions, camels, chame
leons and more. Animal puzzles do not – yet – constitute a mathematical field of their own, but they do provide a delightful and diverse bestiary of brainteasers, allowing me to showcase some of my favourite puzzles from the Middle Ages to the present day.

  After visiting the animal kingdom, we will find ourselves in peril. In real life you may never have been abandoned on an island, trapped in a maze, locked in a room or stuck on death row. In puzzle-land, however, we get into these scrapes all the time, as you will discover in the second chapter, in which the problems are concerned with escape and survival. You will be required to think laterally, logically, and even topologically. Several of these puzzles are based on fascinating discoveries in computer science, in which devising a strategy to get out of jail, say, is analogous to building an algorithm.

  I’ve written this book to share the joy I get from solving problems. A good puzzle will not only stimulate creative thinking, but also spark a sense of wonder and curiosity about the world. I have taken care to choose questions that present the solver with a surprise, or which reveal an interesting pattern or idea. Puzzles are a versatile medium, covering a huge variety of genres, and I hope this book will tickle your brain from all sides.

  The puzzles are not organised by level of difficulty. You can read the chapters from beginning to end or skim them and pick and choose. I include material about the history of mathematics, and the role of puzzles within it, and have full explanations and follow-on discussions in the back. The problems in the ‘Tasty teasers’ sections are snacks to get you in the mood.

  Indeed, you will have nibbled a puzzle already, the one on the cover about the folded shape. I love this puzzle because the first letter that comes to mind when you try to unfold the shape is an L, which is the wrong answer. It takes some mental effort to discard the obvious letter, at which point a flash of insight may reveal the less obvious one. Puzzles often play with our minds this way, deliberately leading us up the garden path, or presenting us with a tantalising, but entirely erroneous, solution. The pleasure in solving a puzzle that is trying to misdirect makes the final ‘aha!’ especially sweet.

  The area of mathematics in which we are most handicapped by our own psychology is probability, the theme of the final chapter. Our brains are poorly equipped to understand randomness, and probability puzzles are a great way to identify where our intuitions go wrong. Not only do these puzzles surprise and enlighten us, they also help us think more clearly.

  Indeed, that is the power of all puzzles. They are fun. But they are also useful. Puzzles make our brains more nimble, versatile, flexible and multifaceted. They improve our capacity for reasoning, hone our ability to spot patterns, train us to look at the world from different perspectives, and point out areas where we are easily misled.

  Now pack your bags.

  The animals are eager to meet you.

  Tasty Teasers

  Number conundrums

  1)

  Each of the three big circles passes through four small white circles. Place the digits from 1 to 6 in the white circles so that the numbers on each big circle add up to 14.

  2)

  Place the digits from 1 to 8 in the circles such that no number is connected by a line to a number either 1 more or 1 less than itself. For instance, 6 cannot be connected to 5 or 7.

  3)

  Divide this clock face with two straight lines so that the sum of the numbers in each section is the same.

  4) Place the digits indicated in the squares so that each equation makes sense. For example, the first one uses the digits 1,2,3 and 4.

  5)

  Place seven digits from 1 to 9 in the circles so that the three digits on each line multiply to the same amount.

  6)

  In the triangle the digits from 1 to 6 are positioned such that the difference between two adjacent numbers is shown in the row beneath them.

  Do the same for the digits 1-10 in the triangle on the left, and the digits 1-15 in the triangle on the right.

  The puzzle zoo

  ANIMAL PROBLEMS

  1

  THE THREE RABBITS

  Can you rearrange the positions of the three rabbits and the three ears so that each rabbit has two ears?

  If you’re reading this in a Devon church, don’t look up!

  More than a dozen churches in Devon have medieval wooden carvings on their ceilings that display the solution. In fact, the image of three rabbits, or hares, sharing their ears is a symbol that appears in many sacred sites across the northern hemisphere, the earliest dating from sixth-century China. The largest cluster of examples, however, is in Devon, where the linked leporine lugs are sometimes known as the Tinners’ Rabbits, possibly because wealth from local tin mines helped build and maintain the churches hundreds of years ago.

  The simple symmetry of the three rabbits makes them a powerful mystical symbol, an easy metaphor for ideas of eternity and beauty. Yet part of their allure is their elegance as a puzzle, in which the image makes sense when seen one way, but not when seen in another. We are attracted to images that play with our sense of perception. Puzzles are by their nature mesmerising; they spin our heads right round.

  2

  DEAD OR ALIVE

  These dogs are dead you well may say:

  Add four lines more, they’ll run away!

  This canine conundrum dates from 1849, when it appeared in the first issue of The Family Friend, a lifestyle magazine in book form aimed at the Victorian housewife. It’s the original version of a visual illusion that has been reinvented many times since, in which you must draw four lines on an image in order to bring two animals back to life.

  Talking of rabbits and dogs…

  3

  GOOD NEIGHBOURS

  A woman opened the door and her dog walked in. The dog had the neighbour’s pet rabbit in its mouth. The rabbit was dead. Distraught, the woman went next door immediately to apologise. The neighbour smiled. ‘Don’t worry, my rabbit was not harmed.’

  Why was the rabbit not harmed?

  The mysticism around the three rabbits also turns on rabbit behaviour in the wild. The breeding rate of bunnies has made them age-old symbols of fertility and rebirth, anthropomorphic shorthand for the highly sexed. Prolific procreation is their principal protection against a profusion of prowling predators. Reproduction is ripe for mathematical analysis. Indeed, one of the most famous problems in mathematical literature is about a rapidly expanding rabbit family.

  In the Liber Abaci, the thirteenth-century book that introduced Arabic numerals to Europe, Leonardo of Pisa, better known as Fibonacci, set the following problem. Start with a pair of rabbits. If they produce a pair of offspring every month, and if every new pair becomes fertile after a month, again producing another pair every month, how many pairs are there at the end of 12 months? Saving you the bother of calculation, the monthly totals are (1), 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 and 377, an ordered list of numbers known – thanks to this problem – as the Fibonacci sequence.

  (We start with 1 pair. At the end of the first month, this pair has bred, so we have 2 pairs. At the end of the second month, the first pair has bred again but the second pair is not yet fertile, so we have a total of 3 pairs. At the end of the third month, the first and second pairs have now bred, but the third is not yet fertile, giving a total of 5 pairs. And so on.)

  The Fibonacci sequence is one of the few number sequences known beyond mathematics. It has many intriguing properties. For example, each term is the sum of the previous two (1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on). This recursive process models natural growth – for plants as well as bucktoothed mammals – which is why the number of spirals in cauliflower, Romanesco broccoli, pinecones, pineapples and sunflower heads is nearly always a number in the Fibonacci sequence. Do check, next time you’re at the supermarket.

  Fibonacci hit on a fascinating piece of pure mathematics. Yet how well does his problem reflect the real world? In other words, did he create an accurate model of
the reproductive practices of rabbits?

  To Fibonacci’s credit, he got the gestation period about right. Rabbits take a month to have babies, and can be impregnated again within minutes of giving birth. Poor things. So, theoretically, a doe can have 12 litters a year. On the other hand, she becomes fertile more slowly, after about six months, and she has more children, on average about six kits per litter. Armed with the relevant zoological data we can update Fibonacci’s historic rabbit riddle.

  4

  A FERTILE FAMILY

  How many descendants does a female rabbit have in her lifetime if:

  Rabbits become fertile after six months.

  Once fertile, a doe produces every month a litter of six kits, three of which are female.

  The life span of a rabbit is seven years.

  Of course, these details still paint a simplified picture. In the wild, the life span of a rabbit is only about a year. After a few years the fertility of female rabbits drops off. Environmental factors such as available space and food will limit the rate of growth. Nevertheless, the question provides a theoretical estimate of the upper bound of potential bunny reproduction, a scientific analysis of what it really means to breed like rabbits.

  I’ll give you full marks if you can work out the formula for how to calculate the answer. To get the exact figure you might need some computer assistance. For those who aren’t masters of technology (or Excel), look in the back. But before you do, estimate what you think the answer is. If you get it right to within two powers of 10 – i.e., up to 100 times more, or 100 times less – treat yourself to a meal of lapin à la moutarde and a bottle of Chablis. You will be amazed.