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 9flips 
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 9flips 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 