How to Solve It

G. Polya's How to Solve It is a guide to the methods and mental operations involved in solving problems, with particular emphasis on mathematical problems. Drawing on a long tradition of heuristic, the study of methods and rules of discovery, Polya argues that problem-solving is a practical skill, like swimming, that can be learned through imitation and practice. The book is organized into four parts: "In the Classroom," a short dialogue, a "Short Dictionary of Heuristic," and a set of problems with hints and solutions.

Polya frames his central motivation at the outset. As a student, he found himself asking not just whether a solution was correct but how anyone could have invented it. This question drove his study of heuristic. He distinguishes two faces of mathematics: the rigorous, deductive science of Euclid and the experimental, inductive science of mathematics "in the making." The second face, he argues, has never before been presented to students in quite the way he attempts.

Part I, "In the Classroom," opens with five sections on the teacher's purpose. Polya argues that the teacher's most important task is helping students develop independent problem-solving ability by calibrating assistance carefully. The teacher should ask questions the student might have asked, such as "What is the unknown?" These questions recur across countless problems, and behind each lies a mental operation useful for problem-solving. The same questions apply to algebraic, geometric, theoretical, and practical problems, with certain questions suiting "problems to find" (seeking an unknown value) and others suiting "problems to prove" (establishing whether an assertion is true or false).

Polya then presents the four phases of problem-solving. The first, "Understanding the problem," requires identifying the unknown, the data, and the condition linking them, as well as drawing a figure and introducing suitable notation. The second, "Devising a plan," is the central achievement; it depends on recollection of related problems or theorems. Key suggestions include "Do you know a related problem?" and "Look at the unknown!" The third, "Carrying out the plan," demands patience and careful verification of each step. The fourth, "Looking back," involves reconsidering the completed solution: checking the result, deriving it differently, and asking whether the method can be used elsewhere.

Polya illustrates all four phases through a running example: finding the diagonal of a rectangular parallelepiped (a box shape) given its three edge lengths. Through guided questioning, the student recalls that finding the side of a right triangle is a related solved problem, introduces an auxiliary right triangle, and applies the Pythagorean theorem twice to derive the formula. In the looking-back phase, Polya demonstrates multiple checks: verifying that all data appear in the formula, confirming symmetry, testing limiting cases, and verifying dimensional consistency.

Polya discusses the teacher's method of questioning, which begins with a general question and descends gradually to more specific suggestions if needed. He contrasts this with a bad question like "Could you apply the theorem of Pythagoras?", which gives the secret away and teaches nothing transferable. Three further worked examples demonstrate the method across problem types: a construction problem (inscribing a square in a triangle), a proof problem (showing that angles with parallel sides pointing in corresponding directions are equal), and a rate problem (finding how fast the water surface rises in a conical vessel).

Part II presents the problem-solving process as a concise dialogue, subdividing the work into five stages. The first phase from Part I splits into "Getting acquainted," where the solver visualizes the problem as a whole, and "Working for better understanding," where the solver isolates principal parts and considers them individually and in combination. "Hunting for the helpful idea" follows, as the solver considers the problem from various angles and pursues even incomplete ideas. "Carrying out the plan" means working through each step from the decisive idea. "Looking back" involves simplifying the solution and looking for further applications.

Part III, the "Short Dictionary of Heuristic," is the most extensive section. Its 67 alphabetically arranged articles illustrate and expand on the concepts underlying the list. A key organizational article, "Modern Heuristic," explains that the discipline draws on logic, psychology, and historical writers such as the ancient Greek mathematician Pappus, the philosopher and mathematician René Descartes, and the mathematician and philosopher Gottfried Wilhelm Leibnitz, as well as on unbiased experience. Polya identifies 12 principal articles that study various aspects of the solving process at length.

The article on "Analogy" defines it as a sort of similarity in which objects agree in certain relations of their parts. Polya demonstrates the power of solving a simpler analogous problem: finding the center of gravity of a tetrahedron (a solid with four triangular faces) by first solving the analogous problem for a triangle. Inference by analogy, he notes, yields plausible conjectures that must be tested but should not be ignored. "Auxiliary Elements" and "Auxiliary Problem" explain how new elements and subordinate problems further a solution, discussing equivalent reduction (where two problems are each solvable through the other) and unilateral reduction (where the solver passes to a more or less ambitious problem). The "Inventor's Paradox" states that the more ambitious plan may have more chances of success, provided it rests on genuine insight.

"Decomposing and Recombining" explains how solvers break a problem into parts and reassemble them. Polya demonstrates the method of intersecting loci: In a geometric construction, keeping only one part of the condition at a time produces a locus (the set of all points satisfying that condition) for each part, and the unknown point lies at their intersection. "Generalization" and "Specialization" show how varying a problem's scope can make it more tractable. Generalization can simplify a problem by isolating its essential property, while specialization is useful for testing conjectures and as a stepping stone to the general case.

"Induction and Mathematical Induction" distinguishes two processes. Induction discovers general laws by observing particular instances; mathematical induction provides rigorous proof by showing a formula holds for a base case and that validity for any value n implies validity for n + 1. Polya argues that mathematics in the making is experimental and inductive, even though finished mathematics is deductive. The article on "Pappus" presents the ancient mathematician's description of analysis and synthesis. In analysis, the solver starts from what is required, assumes it done, and derives antecedents until reaching something known. In synthesis, the solver reverses the process. Analysis corresponds to devising a plan; synthesis to carrying it through.

Other articles address figures, notation, the distinction between problems to find and problems to prove, reductio ad absurdum (proving impossibility by deriving an absurdity from an assumption), indirect proof (proving a claim by assuming its opposite and deriving a contradiction), and heuristic reasoning. Several articles treat attitudes and habits: "Determination, Hope, Success" argues that willpower and emotional engagement are essential; "Routine Problem" warns that limiting students to mechanical exercises destroys inventive faculties. "Why Proofs?" defends the role of proofs in education as tools for developing logical rigor. "Working Backwards" illustrates analysis through a water-measuring puzzle, and "Wisdom of Proverbs" collects folk sayings that characterize each phase of problem-solving, arguing that the essence of heuristic discussion has been preserved in proverbial wisdom since the beginning of human society.

Part IV, "Problems, Hints, Solutions," offers 20 problems requiring no more than high-school-level knowledge but demanding originality and ingenuity. Each is followed by a hint quoting a suggestion from the list and a worked solution that models the reasoning process, demonstrating techniques discussed throughout the book.