An interactive exploration
Four publishers rejected this book. It sold over a million copies. Pólya took the messy, human process of solving problems and made it teachable.
"A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery."
Every problem — crossword puzzle, research theorem, or debugging session — passes through four phases. Click each to explore.
"It is foolish to answer a question that you do not understand." Most failures happen here — rushing into computation without grasping what you're looking for.
→ What is the unknown?
→ What are the data?
→ What is the condition?
→ Draw a figure. Introduce notation.
→ Is it possible to satisfy the condition?
Where the magic happens. You don't invent solutions from nothing — you connect to problems you've already solved. The gap between "stuck" and "eureka" is bridged by systematic exploration.
→ Do you know a related problem?
→ Look at the unknown! Think of a familiar problem with the same unknown.
→ Could you restate the problem?
→ If you can't solve it, solve a simpler related problem first.
The creative work was Phase 2. Phase 3 is patience and rigor. Check each step. The heuristic reasoning that got you here must now give way to strict argument.
→ Check each step.
→ Can you see clearly that the step is correct?
→ Can you prove that it is correct?
The most skipped, most valuable phase. Reviewing consolidates knowledge. One solved problem becomes a tool for solving ten more.
→ Can you check the result?
→ Can you derive the result differently?
→ Can you see it at a glance?
→ Can you use the result, or the method, for some other problem?
Work through real problems using Pólya's four phases as scaffolding. Each phase offers guiding questions and reveal-on-demand hints — not answers.
"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice."
Pólya's toolbox of strategies for when you're stuck. Click a tool to see how it works and when to reach for it.
"No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all."
Pólya's discussion of Pappus: analysis works backwards from the desired result to find the path; synthesis presents the solution forwards. Click nodes to trace reasoning in each direction.
"Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science."
Problem: Inscribe a square in a given triangle (one side of the square on a side of the triangle).
"The more ambitious plan may have more chances of success." Sometimes a more general problem is easier than a specific one, because the general structure reveals the mechanism.
"If you will sail without danger, you must never put to sea."
Example: Sum of cubes. Can you find a formula for 1³ + 2³ + 3³ + ... + n³?
The single case gives you a number. The general approach gives you the pattern. Generalizing made the problem easier, not harder.
Pólya's most actionable advice: focus on what you're trying to find, then recall problems with the same type of unknown. This narrows the search space from "all of mathematics" to a tractable set.
← Select an unknown type
Pólya's core pedagogical insight: help the student, but not too much. The teacher asks questions that could have occurred to the student. Play the student — choose your responses.
"If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity."
Scenario: Find the diagonal of a rectangular box with sides a, b, c.
Pólya's book contains 67 alphabetically arranged articles — a "Short Dictionary of Heuristic." Here are the standout entries.
"Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves." — Leibniz
This book is a tool, not a decoration. Pólya didn't write a philosophy of problem-solving — he wrote a manual. The four phases are obvious once stated, which is the hallmark of a great insight. You can't unsee them.
What surprised me most: how much Pólya emphasizes emotion. Determination, hope, the thrill of a bright idea, the dejection of being stuck. Problem-solving is not a cold logical exercise — it's a deeply human struggle.
"The first rule of discovery is to have brains and good luck. The second rule is to sit tight and wait till you get a bright idea."
What dates it: Gendered language, deep Euclidean assumptions. What doesn't: Everything else. The framework works for debugging code, writing essays, or designing systems — not just geometric proofs.
Who should read it: Everyone who solves problems — which is everyone. Especially teachers, students, engineers, anyone who has ever been stuck and wanted a systematic (but not mechanical) way to get unstuck.
"Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice."
Read this book. Then solve something hard.