• Title Page (3)
• Copyright Page (4)
• Dedication (6)
• Foreword (7)
• Contents (9)
• 1. Jugs and Stamps: How To Solve Problems (12)
• 1.1. Introduction (12)
• 1.2. A Drinking Problem (13)
• 1.3. About Solving Problems (14)
• 1.4. Rethinking Drinking (19)
• 1.5. Summing It Up (21)
• 1.6. Licking a Stamp Problem (25)
• 1.7. A Little Explanation (29)
• 1.8. Tidying Up (30)
• 1.9. Generalise (32)
• 1.10. In Conclusion (35)
• 1.11. Epilogue (38)
• 1.12. Solutions (39)
• 2. Combinatorics I (51)
• 2.1. Introduction (51)
• 2.2. What is Combinatorics? (51)
• 2.3. The Pigeonhole Principle (54)
• 2.4. Counting without Counting (65)
• 2.5. A Sigma Aside (81)
• 2.6. Solutions (83)
• 3. Graph Theory (101)
• 3.1. Introduction (101)
• 3.2. Konigsberg (101)
• 3.3. So What is a Graph? (107)
• 3.4. Ramsey (116)
• 3.5. Euler Tours ⠀刀攀瘀椀猀椀琀攀搀) (120)
• 3.6. Knight's Tours (121)
• 3.7. Hamilton (125)
• 3.8. Trees (129)
• 3.9. Planarity (130)
• 3.10. The Four Colour Theorem (136)
• 3.11. Some Additional Problems (146)
• 3.12. Solutions (148)
• 4. Number Theory 1 (169)
• 4.1. What is It? (169)
• 4.2. Divisibility by Small Numbers (172)
• 4.3. Common Factors (181)
• 4.4. Fermat's Little Theorem (191)
• 4.5. A.P.'s (196)
• 4.6. Some More Problems (201)
• 4.7. Solutions (203)
• 5. Geometry 1 (224)
• 5.1. Introduction (224)
• 5.2. Squares (224)
• 5.3. Rectangles and Parallelograms (231)
• 5.4. Triangles (234)
• 5.5. Circles (239)
• 5.6. Solutions (247)
• 6. Proof (274)
• 6.1. Introduction (274)
• 6.2. Why Proof? (274)
• 6.3. Proof by Contradiction (279)
• 6.4. Mathematical Induction (282)
• 6.5. Conclusion (294)
• 6.6. Solutions (296)
• 7. Geometry 2 (312)
• 7.1. Cartesian Geometry (312)
• 7.2. Lines (312)
• 7.3. Modulus (319)
• 7.4. Loci: One Fixed Point (326)
• 7.5. The Cosine Rule (329)
• 7.6. Loci: Two Points (332)
• 7.7. Conics (341)
• 7.8. Solutions (342)
• 8. Some IMO Problems (368)
• 8.1. Introduction (368)
• 8.2. What is the IMO? (368)
• 8.3. PHIL 1 (370)
• 8.4. MON 1 (372)
• 8.5. MON 6 (374)
• 8.6. UNK 2 (375)
• 8.7. Hints — PHIL 1 (376)
• 8.8. Hints — MON 1 (379)
• 8.9. Hints — MON 6 (381)
• 8.10. Hints — UNK 2 (383)
• 8.11. Solutions (386)
• Index (407)