The Art of Problem Solving

About the book

The Art of Problem Solving, by Richard Rusczyk and Sandor Lehoczky, treats mathematics as a set of techniques for attacking problems. The material grew out of competition training and covers topics outside the standard secondary sequence, so readers meet number theory, combinatorics and inequalities alongside algebra and geometry. The emphasis falls on method: how to read a problem, choose a strategy and check an answer.

The book works as a course rather than a reference. Explanations lead into worked examples, and the exercises carry much of the teaching load. Many problems come from contests, which gives the practice a particular flavor: compact statements, little scaffolding and a premium on finding the right first move. Solutions and hints support independent study.

Both authors have competition backgrounds, and that audience shapes the tone. The prose stays lean and expects effort. Readers who want a gentle tour of school mathematics will find the pace brisk and the problem sets demanding.

How to read it

Work through the examples with paper in hand. The text often states a technique and demonstrates it once; the reasoning becomes clear when you try the step yourself before reading on. Skipping the exercises turns a course into a list of facts.

Expect uneven difficulty across topics. Some sections move quickly through material that assumes comfort with algebra, while others slow down for less familiar ideas. When a problem resists, consult the solutions, then return to a similar exercise and attempt it unaided. Pace yourself across weeks rather than days.

Who it suits

Students preparing for mathematics competitions will find targeted practice and techniques they can apply under time pressure. Strong middle and high school students who enjoy puzzles and want to go beyond their school syllabus get a structured path. Teachers and coaches can use the problem sets as material for clubs and enrichment sessions. Adults returning to mathematics for pleasure may also find the approach rewarding if they are willing to struggle with hard problems.

Reading considerations

The book assumes fluency with basic algebra and a tolerance for frustration; it is not an introductory text for readers who need step-by-step remediation. Volume arrangements vary between editions, so check which volume and printing you have before planning a sequence, since exercise numbering and coverage can differ. The competition focus means some topics receive more attention than a standard curriculum would give them.