Master the Art of Problem Solving: Polya’s Four Steps and Nine Strategies

1. Understanding the Problem: The Starting Point

Problem solving begins with thinking, not acting. We must fully grasp the problem’s requirements, knowns, and unknowns.

1. Core Questions

When facing a problem, ask:

What is the unknown?

What known conditions are given?

How might these conditions be related?

Many failures stem from not clarifying the actual question.

2. Example Illustration

Example: A right‑triangle has area 25 m²; find the length of its hypotenuse expressed as a function of the perimeter.

The first step is understanding the problem.

Known: Area = 25 m²

Requirement: Express hypotenuse as a function of perimeter

Implicit condition: The triangle is right‑angled

Let the legs be … and the hypotenuse …

2. Planning: Core Strategies

After understanding, choose appropriate tools and strategies to bridge knowns and unknowns. Polya’s strategies are not fixed answers but thinking paths; nine common strategies are listed.

1. Find Familiar Content

When encountering a new problem, look for structurally or logically similar problems; learning transfers familiar patterns.

Example: In a function problem, a graph resembling … suggests trigonometric properties.

2. Discover Patterns

Some problems are solved by inductively finding patterns in sequences, shapes, or variables.

Example: A sequence … reveals a triangular‑number pattern with formula …

3. Analogy and Mapping

Simplify complex problems into more understandable versions.

High‑dimensional → low‑dimensional

Abstract → concrete

General → specific

Example: To compute a solid’s volume, first consider the area of a cross‑section.

4. Introduce Auxiliary Quantities

Draw auxiliary lines in geometry or introduce new variables in algebra.

Example: For a trapezoid’s area, complete it into a rectangle or two triangles.

5. Case Analysis

Essential for absolute values, sign functions, piecewise definitions.

Example: Solve an inequality by discussing three cases based on the absolute‑value definition.

6. Backward Reasoning

Start from the desired conclusion and work backward toward known conditions.

Example: Solve an equation by setting the target and retracing steps.

7. Set Sub‑Goals

Break complex problems into intermediate objectives.

Example: In a series sum, first prove a recurrence, then use induction.

8. Indirect Reasoning

Includes proof by contradiction or contrapositive.

Example: To prove “if n is even then n² is even,” assume n is odd and reach a contradiction.

9. Mathematical Induction

A powerful tool for recursive problems.

Steps:

Verify the base case.

Assume the statement holds for k.

Prove it holds for k + 1.

3. Executing the Plan: Implement Each Step

After planning, carry out each step in order.

Key Points

Each step must have clear logical justification.

Use precise mathematical language (“let”, “then”, “because”, “therefore”).

Avoid leaps; facilitate checking and review.

Continue the earlier example of the right‑triangle with area 25 m².

Solution Process

Let the legs be a and b, the hypotenuse c, and the perimeter P.

Establish equations: area = (a·b)/2 = 25, a² + b² = c², P = a + b + c.

Use the Pythagorean theorem and area relation to express c as a function of P.

The resulting expression gives the required function.

4. Review and Reflection: The Golden Loop

After solving, the most valuable yet often ignored step is reflection.

Four Review Questions

Are there calculation errors?

Is there a simpler solution?

Can the method be analogized to other problems?

Can the problem be solved again using a different approach?

“I find that every problem I solve often inspires solutions to other problems.” – René Descartes, Discours de la méthode

This illustrates how problem‑solving knowledge can be distilled, recombined, and applied across domains.