Problem Solving Strategies - A Complete Toolkit
Every mathematician – and every student – eventually faces a problem they do not immediately know how to solve. The difference between someone who gives up and someone who succeeds is not talent alone: it is having a toolkit of strategies and the patience to apply them systematically. This page gives you that toolkit.
Working backwards from a goal is a far older idea than it might seem. The Greek mathematician Pappus of Alexandria, writing around 320 CE, described a technique he called the “method of analysis”: start by assuming the thing you want to prove is already true, then work backwards through a chain of logical consequences until you reach something already known to be true (or false, which tells you the original assumption was wrong). This is the direct ancestor of the “work backwards” strategy taught below, and it shows that systematic problem-solving heuristics have been studied for well over a thousand years before Polya wrote them down in a modern form.
George Polya’s Four-Step Method
In 1945, Hungarian mathematician George Polya published How to Solve It – one of the best-selling mathematics books of all time. He proposed four universal steps for tackling any problem:
1. Understand the problem. What is given? What is unknown? Can you restate it in your own words?
2. Devise a plan. Which strategy might work? Have you seen a similar problem?
3. Carry out the plan. Execute each step carefully, checking as you go.
4. Look back. Is the answer reasonable? Can you check it a different way? Could this method solve similar problems?
Strategy 1: Draw a Diagram or Table
Many problems become obvious once visualised. Drawing a diagram externalises the problem – it moves information from your head to paper where it can be seen and manipulated. Tables organise information so patterns become visible.
Draw 6 dots and connect every pair. Count the lines, or notice each person shakes 5 hands: total = (6 × 5) / 2 = 15 handshakes.
Dividing by 2 avoids counting each handshake twice.
Strategy 2: Look for a Pattern
Calculate several small cases, record them in a table, and look for a repeating rule. Once you spot the pattern, express it as a formula and verify it.
List units digits: 71=7, 72=9, 73=3, 74=1, 75=7, ... → repeating cycle of 4.
2026 ÷ 4 = 506 remainder 2. The units digit is the same as 72, which is 9.
Strategy 3: Work Backwards
Start from the desired answer and reverse the operations to find the starting conditions. Especially useful when the end state is known but the starting point is not.
Work back: before the £15 spend → £25 + £15 = £40. Before halving → £40 × 2 = £80.
Strategy 4: Consider a Simpler Case
Replace large numbers or complex conditions with the simplest possible version. Solve the simpler problem, understand why it works, then scale the method up.
Simplify first: what is 1 + 2 + 3 + 4 + 5? Pair them: (1+5) + (2+4) + 3 = 6 + 6 + 3 = 15.
Pattern: n(n+1)/2. For n = 100: 100 × 101 / 2 = 5 050.
Strategy 5: Break the Problem into Sub-Problems
A complex problem is often a collection of simpler problems joined together. Identify the sub-problems, solve each one, and combine the results.
Split into two rectangles, or use subtraction:
Total rectangle: 8 × 6 = 48 m². Removed piece: 3 × 2 = 6 m².
L-shape area = 48 − 6 = 42 m².
Strategy 6: Use Algebra
Name the unknown quantity with a variable, translate the problem conditions into an equation, then solve. This strategy converts a wordy problem into a precise mathematical one.
Let the integers be n, n+1, n+2.
n + (n+1) + (n+2) = 87 → 3n + 3 = 87 → 3n = 84 → n = 28.
The integers are 28, 29, and 30.
Strategy 7: Guess, Check, and Refine
Make an educated guess, check whether it works, then adjust intelligently based on how far off you are. Repeat until the answer is found. This is not random guessing – each check informs the next estimate.
Strategy 8: Change Representation
If a problem is hard to solve in its current form, translate it into a different form: a graph, a number line, coordinates, a matrix, or a percentage instead of a fraction. Sometimes the answer is obvious in one representation but hidden in another.
Common Problem-Solving Mistakes
| Mistake | Better Approach |
|---|---|
| Jumping to a method without understanding the problem | Spend time reading carefully; restate the problem in your own words |
| Giving up after one failed strategy | Systematically try a different strategy from the list |
| Not checking the answer | Always substitute back or use a different method to verify |
| Forgetting units | Carry units through every step and state them in the answer |
Key Takeaways
- Polya’s four steps: Understand, Plan, Execute, Look Back.
- Eight core strategies: diagram, pattern, work backwards, simpler case, sub-problems, algebra, guess-check-refine, change representation.
- No single strategy works on every problem – the skill is choosing the right one.
- Always check your answer by a different method or by substituting back.