Strategies of Problem Solving

Note: the second edition of this book is now available. Solving mathematical problems is both a science and an art. It is a science because we need to learn some basic concepts and skills, and use proper terminology whenexplaining our solution to other people. It is also an art because very often we need to be creative. There are infinitely many types of math problems, and it is impossible to learn how to solve every problem in the world. However, there are a few basic principles that...
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Note: the second edition of this book is now available. Solving mathematical problems is both a science and an art. It is a science because we need to learn some basic concepts and skills, and use proper terminology whenexplaining our solution to other people. It is also an art because very often we need to be creative. There are infinitely many types of math problems, and it is impossible to learn how to solve every problem in the world. However, there are a few basic principles that are good to know. There are a few approaches and methods that are often useful. In this book, we discuss some of them, including various types of proofs, the pigeon hole principle, the principle of mathematical induction, invariants, coloring, etc. In each chapter, we provide basic definitions and facts to get you started. We do not prove most of the facts given in this book, since our main goal is to learn howto solve problems, i.e. use these facts. They are usually proved in other college courses such as abstract algebra and analysis. Sometimes, however, the idea of a proof of a theorem can be used for solving many problems. In such cases we provide the proof.The book contains over 300 problems on various topics and detailed solutions of approximately half of them. This book is primarily intended for college students and mathematics teachers. Most chapters are accessible to middle and high school students as well. It would especially be helpful for those competing in mathematics contests and wishing to improve their problem solving skills.