How to Solve It
Problem-solving develops through understanding, planning, execution, and review rather than waiting for an unexplained flash of genius.
Understanding requires identifying the unknown and the given conditions.
Being stuck does not mean lacking mathematical talent. Polya makes the next questions practical: what is given, what is missing, and whether a simpler related problem can reveal a route.
A useful representation can reveal relationships hidden in wording.
Planning draws on related problems and previous methods.
A simpler case may expose the structure of a difficult one.
Working backward can connect a desired result with known information.
Looking back turns an answer into learning. A successful solution can still hide a fragile assumption, while an unsuccessful attempt may contain an approach worth carrying into another problem.
Execution requires checking each step rather than trusting the plan blindly.
Failure can reveal that an assumption or approach needs revision.
Looking back tests the answer and extracts a reusable idea.
Heuristics guide inquiry without guaranteeing success.
The conclusion presents mathematical ingenuity as a practice that can be cultivated.