Problem-Solving Through Problems

1. Front Matter

   Pages i-xi

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2. Heuristics

   • Loren C. Larson

   Pages 1-57

3. Two Important Principles: Induction and Pigeonhole

   • Loren C. Larson

   Pages 58-83

4. Arithmetic

   • Loren C. Larson

   Pages 84-119

5. Algebra

   • Loren C. Larson

   Pages 120-153

6. Summation of Series

   • Loren C. Larson

   Pages 154-191

7. Intermediate Real Analysis

   • Loren C. Larson

   Pages 192-240

8. Inequalities

   • Loren C. Larson

   Pages 241-279

9. Geometry

   • Loren C. Larson

   Pages 280-316

10. Back Matter

    Pages 317-333

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The purpose of this book is to isolate and draw attention to the most important problem-solving techniques typically encountered in undergradu­ ate mathematics and to illustrate their use by interesting examples and problems not easily found in other sources. Each section features a single idea, the power and versatility of which is demonstrated in the examples and reinforced in the problems. The book serves as an introduction and guide to the problems literature (e.g., as found in the problems sections of undergraduate mathematics journals) and as an easily accessed reference of essential knowledge for students and teachers of mathematics. The book is both an anthology of problems and a manual of instruction. It contains over 700 problems, over one-third of which are worked in detail. Each problem is chosen for its natural appeal and beauty, but primarily to provide the context for illustrating a given problem-solving method. The aim throughout is to show how a basic set of simple techniques can be applied in diverse ways to solve an enormous variety of problems. Whenever possible, problems within sections are chosen to cut across expected course boundaries and to thereby strengthen the evidence that a single intuition is capable of broad application. Each section concludes with "Additional Examples" that point to other contexts where the technique is appropriate.

• Arithmetic
• Cauchy-Schwarz Inequality
• Derivative
• Division
• Problem solving
• Schwarz inequality
• calculus
• fundamental theorem
• mean value theorem
• pigeonhole principle

From the reviews:

"This is a very welcome addition. The main message of the book is that the only way to learn to solve problems is to solve problems! I found this book very helpful. I am quite sure the book will be in constant use and I have no hesitation in recommending it." (The Mathematical Gazette)

• Department of Mathematics, St. Olaf College, Northfield, USA

  Loren C. Larson

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