How to Solve It: Pólya's Method

Most people, faced with a hard problem, do one of two things: they freeze, or they lunge at the first method that comes to mind. Neither works reliably. What separates steady problem-solvers is not raw cleverness but a habit of moving through the problem in a deliberate order — and that order can be taught.

In 1945 the Hungarian-American mathematician George Pólya distilled that order into four plain phases. His little book How to Solve It has never gone out of print, and its influence reaches far beyond mathematics into engineering, programming, and everyday reasoning. The value is that the phases are simple enough to remember under pressure yet general enough to fit almost any problem you meet.

The four phases

Pólya named four phases. First, understand the problem: what is the unknown, what are the data, what is the condition connecting them? Second, devise a plan: find the link between what you have and what you want, often by recalling a related problem. Third, carry out the plan: execute each step, checking that each one is correct. Fourth, look back: examine the solution, test it, and ask whether the result or the method could be used again. The phases are a loop, not a staircase — understanding often deepens while you plan, and a failed plan sends you back a step.

Heuristics: the questions that unstick you

Pólya's real gift was a catalogue of heuristics — questions and moves that generate ideas when you are stuck. Have you seen a related problem? Can you restate it? Could you solve a simpler or more special version first? Can you work backwards from the goal? Can you check the result by a different method? These are not guarantees; they are prompts that reliably shake loose a plan when none is obvious.

Why 'look back' is the phase people skip

Once an answer appears, relief pulls us straight to the next task. But the look-back phase is where an answer becomes knowledge. Checking the result catches errors; re-deriving it a second way builds confidence; asking 'where else does this method apply?' turns one solved problem into a reusable technique. Skipping it means solving the same kind of problem slowly forever.

Problem: the sum of two numbers is 20 and their difference is 4. Find them. Understand: the unknowns are two numbers; the data are their sum and difference. Plan: try the heuristic 'work backwards from the relationships' — if the larger number were reduced by the difference, both would be equal, so twice the smaller number is 20 − 4 = 16. Carry out: the smaller number is 8, so the larger is 12. Look back: check — 8 + 12 = 20 and 12 − 8 = 4, both conditions hold. Then ask: this trick of removing the difference works for any sum-and-difference pair, so I have a reusable method, not just an answer.

Suppose you skip understanding and jump to a plan. Given 'a bat and a ball cost $1.10; the bat costs $1.00 more than the ball,' the fast plan says the ball is $0.10. But the look-back check fails: that makes the bat $1.10 and the total $1.20. The condition 'a dollar more' was never properly understood. Reading the condition carefully — bat = ball + $1.00 — gives ball = $0.05. The method did not fail; a skipped phase did.

George Pólya (1887–1985) was a mathematician at Stanford who cared deeply about how discoveries are actually made, not just how they are written up afterwards. How to Solve It was first published by Princeton University Press in 1945. In it he laid out the four phases and a long dictionary of heuristics drawn from his own teaching of mathematics.

The book's reach turned out to be enormous and verifiable. It has sold well over a million copies and been translated into many languages. Its influence is explicit in computer science: early artificial-intelligence researchers, including work associated with Allen Newell and Herbert Simon, drew on the idea of heuristic search, and countless programming and engineering curricula still teach 'understand, plan, implement, review' as a direct descendant. Pólya's insight was that problem-solving has a describable structure — and that naming it helps ordinary people do it better.

• The four phases must run strictly in order. They form a loop; you revisit earlier phases constantly.
• Heuristics guarantee a solution. They generate ideas; they do not promise one.
• Understanding is the quick part. It is where most failures begin, and deserves the most patience.
• Once you have the answer, you are done. Looking back is what converts an answer into a method you keep.

Take any problem you are chewing on and, out loud, say which of the four phases you are in right now. If you cannot honestly say you have finished 'understand the problem,' you are not allowed to start planning. Then pick one heuristic — restate it, simplify it, or work backwards — and try only that. Notice how naming the phase slows the lunge and steadies the work.

Think Like a Maester: Before hunting for a plan, make sure you can state exactly what is unknown and what connects it to what you know.

1. Name Pólya's four phases in order and what each is for.
2. Why does he treat the phases as a loop rather than a fixed sequence?
3. Give two heuristics from the book and describe when each helps.
4. Why is 'look back' the phase people most often skip, and what does skipping it cost?
5. In the bat-and-ball example, which phase was neglected, and how did that produce the wrong answer?

George Pólya's 1945 classic How to Solve It gave problem-solving a teachable shape: understand the problem, devise a plan, carry out the plan, and look back — a loop rather than a staircase — supported by heuristics such as restating the problem, solving a simpler version, and working backwards. The phases are simple enough to hold under pressure and general enough to fit far beyond mathematics, which is why the book still shapes how engineers and programmers work. The habit that pays most is the one most people drop: looking back, so that each solved problem leaves you with a method, not just an answer.

• What Is a Problem?
• Apollo 13: Problem-Solving Under Pressure