08-17-2026, 02:26 PM
Problem-Solving Through Problems
Author: Loren C. Larson
First published: 1983
Publisher: Springer-Verlag, New York
Series:Problem Books in Mathematics
Loren C. Larson’s Problem-Solving Through Problems is a classic introduction to the art of mathematical problem solving. Rather than organizing mathematics primarily around theories and definitions, Larson organizes the book around techniques that repeatedly appear when attacking unfamiliar problems. The opening chapter develops heuristics such as searching for patterns, drawing diagrams, reformulating a problem, exploiting symmetry, working backwards, considering extreme cases, using contradiction, and generalizing. These strategies are then put to work across induction and the pigeonhole principle, number theory and arithmetic, algebra, series, real analysis, inequalities, and geometry.
The book's strength is its emphasis on learning mathematics by actually solving problems. It contains more than 700 problems, with over one-third worked out in detail, and Larson deliberately selects problems that demonstrate how a relatively small collection of ideas can work in surprisingly different mathematical settings.
The result is somewhere between a textbook, a problem collection, and a handbook of mathematical strategies. It is especially useful for strong undergraduate students, mathematics teachers, and students interested in competitions such as the Putnam. More broadly, the book teaches an important lesson: becoming a better problem solver is less about accumulating isolated tricks and more about developing a repertoire of flexible ideas—and learning to recognize when a familiar idea can unlock an unfamiliar problem.
Key takeaways
Springer — Problem-Solving Through Problems
Author: Loren C. Larson
First published: 1983
Publisher: Springer-Verlag, New York
Series:Problem Books in Mathematics
Loren C. Larson’s Problem-Solving Through Problems is a classic introduction to the art of mathematical problem solving. Rather than organizing mathematics primarily around theories and definitions, Larson organizes the book around techniques that repeatedly appear when attacking unfamiliar problems. The opening chapter develops heuristics such as searching for patterns, drawing diagrams, reformulating a problem, exploiting symmetry, working backwards, considering extreme cases, using contradiction, and generalizing. These strategies are then put to work across induction and the pigeonhole principle, number theory and arithmetic, algebra, series, real analysis, inequalities, and geometry.
The book's strength is its emphasis on learning mathematics by actually solving problems. It contains more than 700 problems, with over one-third worked out in detail, and Larson deliberately selects problems that demonstrate how a relatively small collection of ideas can work in surprisingly different mathematical settings.
The result is somewhere between a textbook, a problem collection, and a handbook of mathematical strategies. It is especially useful for strong undergraduate students, mathematics teachers, and students interested in competitions such as the Putnam. More broadly, the book teaches an important lesson: becoming a better problem solver is less about accumulating isolated tricks and more about developing a repertoire of flexible ideas—and learning to recognize when a familiar idea can unlock an unfamiliar problem.
Key takeaways
- Problem solving is learned through practice: the problems themselves are the central teaching mechanism.
- Heuristics matter: symmetry, parity, contradiction, extreme cases, pattern recognition, reformulation, and working backwards become reusable tools.
- Techniques cross subject boundaries: the same idea may solve problems in algebra, number theory, analysis, or geometry.
- Excellent preparation for advanced problem solving: particularly valuable for undergraduates, teachers, and mathematical competition students.
Springer — Problem-Solving Through Problems
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│ KONSTANTINOS MICHAILIDIS │
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