Литагент HarperCollins: The Ultimate Mathematical Challenge: Over 365 puzzles to test your wits and excite your mind

Copyright

HarperCollins Publishers

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First published by HarperCollins Publishers 2018

FIRST EDITION

© The UK Mathematics Trust 2018

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Source ISBN: 9780008316402

Ebook Edition © November 2018 ISBN: 9780008316419

Version 2018-11-22

Contents

Cover

Title Page

Copyright

Foreword by Marcus du Sautoy

Introduction

The Problems

Week 1

Week 2

Logic Challenge 1

Week 3

Week 4

Crossnumber 1

Week 5

Week 6

Make a Number Challenge

Week 7

Week 8

Crossnumber 2

Week 9

Week 10

Shuttle Challenge 1

Week 11

Week 12

Crossnumber 3

Week 13

Week 14

Logic Challenge 2

Week 15

Week 16

Crossnumber 4

Week 17

Week 18

Shuttle Challenge 2

Week 19

Week 20

Crossnumber 5

Week 21

Week 22

Logic Challenge 3

Week 23

Week 24

Crossnumber 6

Week 25

Week 26

Shuttle Challenge 3

Week 27

Week 28

Crossnumber 7

Week 29

Week 30

Logic Challenge 4

Week 31

Week 32

Crossnumber 8

Week 33

Week 34

Shuttle Challenge 4

Week 35

Week 36

Crossnumber 9

Week 37

Week 38

Logic Challenge 5

Week 39

Week 40

Crossnumber 10

Week 41

Week 42

Logic Challenge 6

Week 43

Week 44

Crossnumber 11

Week 45

Week 46

Crossnumber 12

Week 47

Week 48

The Penultimate Challenge

Week 49

Week 50

Crossnumber 13

Week 51

Week 52

The Ultimate Challenge

Week 53

Solutions

Problems 1 – 366

Make a Number Challenge

Crossnumbers

Logic Challenges

Shuttle Challenges

The Penultimate Challenge

The Ultimate Challenge

Glossary with Some Useful Facts

Table of Squares, Primes etc.

Acknowledgements

About the Publisher

Foreword

Marcus du Sautoy

There are two things that made me fall in love with mathematics. The first was my teacher at my comprehensive school revealing that there was more to mathematics than long division. Seeing the big stories of mathematics about prime numbers, four-dimensional geometry, symmetry and more made me realise the beauty and creativity that bubbles throughout the subject.

The second was when my teacher introduced me to the puzzles of Martin Gardner in Scientific American . It was then that I got hooked on the amazing buzz you get when you’ve struggled to crack a challenging puzzle and then suddenly you see a clever way to unlock the enigma. Be warned: cracking puzzles releases a very addictive drug.

In this book you can immerse yourself in the joys of the second reason I fell in love with mathematics. Jam-packed with problems that have formed part of the UKMT competitions over the years, these challenges range from those aimed at students taking their first steps onto the mathematical terrain to problems that are for those scaling some of the peaks of the subject.

Although the questions formed part of various competitions, I think it is important to remember that mathematics is not at its heart a competitive sport. As mathematicians we all work together to advance our understanding of the universe of numbers and geometry. Every theorem that we prove relies on all the theorems and proofs that previous generations have laid down right back to the Ancient Greeks. Of all the sciences, mathematics is perhaps unique in the way we truly stand on the shoulders of giants to see further and deeper into our subject.

So the only competitive aspect of the problems for you as a reader is the fun of competing against yourself. Enjoy the problems you can’t solve so easily. They are ultimately the ones that will give you the biggest buzz when you crack them. What gets me up in the morning to go to my desk to do mathematics is all the problems I can’t solve. And when you’ve mastered all the challenges of this book, just remember that mathematics has a whole host of big stories that are still waiting for someone to write the final chapter and reveal the mysteries that still obsess us as mathematicians.

Marcus du Sautoy is Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of Oxford.

Introduction

This book contains a selection of problems drawn from the various competitions and other activities organised by the United Kingdom Mathematics Trust (UKMT).

The UKMT was founded in 1996 by bringing together a number of mathematics competitions for school students organised by three different bodies: the British Mathematical Olympiad Committee, the National Committee for Mathematics Contests and the United Kingdom Mathematics Foundation. Today the UKMT organises a range of different competitions and other activities for students of all ages from 10 to 18, with the stated aim ‘to advance the education of children and young people in mathematics’.

This book consists of one short problem for each day of the year, with an interlude every fourteen days that consists of a longer challenge. The difficulty of the problems varies. We have listed at the end of the book the source of each problem. Together with the description of the competition from which it is drawn, this should give the reader some indication of its intended difficulty. The difficulty of a problem is hard to assess and will naturally depend on the reader. Our intention is that the problems should get gradually harder as each week progresses, and from the start of the year to the end. However, you will probably find many exceptions to this general rule.

Many of the problems were originally set as multiple-choice questions but have been rewritten to remove this feature. This may make some of the problems harder than they originally were, although it should be remembered that in all our competitions students are given only a limited amount of time. It is this time pressure, from which the reader of this book is free, that makes our competitions a tough challenge for school students.

In the Olympiad papers students are asked to write out full solutions and not just give the answer. In marking these papers most of the credit is given for coming up with a cogent and clearly expressed argument; a correct numerical answer by itself counts for little. This feature is missing in the Olympiad problems given in this book, as we are not able to mark your answers! For this reason the full flavour of these problems as presented here is missing. You will find it in the UKMT publications, listed below, which cover the Olympiad problems.