| Introduction | xiii |
| Part 1: Short investigations | 1 |
| Advice to the reader | 2 |
| 1 Analysing games | 3 |
| 1.1 Introduction | 3 |
| 1.2 First short investigation | 5 |
| 1.3 Further problems | 8 |
| 1.4 Solutions to exercises in 1.2 | 10 |
| 2 From rhyme to reason | 12 |
| 2.1 Introduction | 12 |
| 2.2 Second short investigation | 12 |
| 2.3 Further problems | 16 |
| 2.4 Solutions to exercises in 2.2 | 21 |
| 3 Thinking things through | 23 |
| 3.1 Introduction | 23 |
| 3.2 Third short investigation | 24 |
| 3.3 Further problems | 29 |
| 3.4 Solutions to exercises in 3.2 | 31 |
| 4 Asking questions | 35 |
| 4.1 Introduction | 35 |
| 4.2 Fourth short investigation | 36 |
| 4.3 Further problems | 47 |
| 4.4 Solutions to exercises in 4.2 | 51 |
| Part 2: Extended investigations | 55 |
| Advice to the reader | 57 |
| Investigation I: Flips | 61 |
| 5 Introducing 9-flips | 63 |
| Solutions | 68 |
| 6 The art of guessing | 69 |
| Solutions | 75 |
| 7 Other flips | 77 |
| Solutions | 80 |
| 8 Flips in other bases | 83 |
| Solutions | 85 |
| 9 Counting 9-flips with n digits | 88 |
| Solutions | 101 |
| 10 Recurrence relations | 103 |
| Solutions | 111 |
| 11 The end of the road | 113 |
| Solutions | 115 |
| Postscript | 117 |
| Investigation II: The postage stamp problem | 119 |
| 12 Getting stuck in | 121 |
| Solutions | 125 |
| 13 Coming unstuck: the search for proof | 127 |
| Solutions | 132 |
| 14 One picture is worth a thousand words | 136 |
| Solutions | 147 |
| 15 The final licking | 152 |
| Solutions | 159 |
| 16 The coin problem | 162 |
| Solutions | 165 |
| Postscript | 167 |
| Solutions | 169 |
| 17 Further problems for extended investigation | 172 |
| 17.1 Introduction | 172 |
| 17.2 Further problems | 175 |
| 17.3 Outlines and hints | 176 |
| Index | 205 |