Problem Solving

A hard problem is rarely beaten by cleverness. It is beaten by a method: understand it, pick a strategy, carry it out, then check.

"I spent half my money, then $6 more, and I have $10 left." How much did I start with?

There is only one number you were actually given, and it is the last one.

"I spent half my money, then $6, and have $10 left" started with spent half spent $6 ? $10 left the only number you are given is the last one

So start at the end and undo. Before spending the $6 I must have had 10 + 6 = $16.

And that $16 was the half that survived. So the whole was $32.

Check it forwards: half of 32 is 16, take $6 and you have $10. It fits.

The method is the point, not the answer. Understand, plan, carry out, look back — and "look back" is the step people skip and the step that catches almost everything.

Undoing a chain of operations in reverse n3n×33n−4−410÷2 forward: ×3, then −4, then ÷2 backward: ×2, then +4, then ÷3 — undo in reverse order n = 8

The four steps

1. Understand. What are you given, and what exactly is being asked? Say it in your own words.

2. Plan. Pick a strategy. Which one is the subject of this whole unit.

3. Carry it out. Do the arithmetic carefully, one line at a time.

4. Look back. Is the answer sensible? Does it answer the question that was asked?

Step 4 is not optional

Most lost marks are not from bad thinking. They are from answering a slightly different question, or from an answer that is obviously too big. Re-reading the question at the end catches both.

Draw a picture

A diagram turns words into something you can look at. Rectangles for areas, a number line for distances, boxes and arrows for stages.

It does not need to be beautiful or to scale. It needs to be labelled.

Work backwards

When a problem tells you the ending and asks for the beginning, run the steps in reverse and undo each one.

Someone spends half their money, then $6, and has $10 left. Undo: before the $6 they had $16; that was half, so they started with $32.

Undo addition with subtraction, and doubling with halving. Always check by running it forwards.

The signal

If the question ends "...how much did they start with?" or "...what was the original number?", work backwards.

Make a table, find a pattern

When something repeats or grows, tabulate the first few cases and look at the differences.

Constant difference means a rule like an + b. A constant ratio means repeated multiplying.

Always test your rule on a case you did not use to build it.

Try a simpler problem

Stuck on 100 people shaking hands? Do 3 people, then 4, then 5. Write the results in a table and the pattern appears.

This is not giving up on the hard problem. It is the standard way mathematicians attack one.

Guess, check, revise

A structured guess beats staring at the page. Guess, test, and note whether you were too high or too low — then use that to aim the next guess.

Three or four rounds usually lands it, and often the pattern in your guesses reveals the equation you could have written.

Worked examples

Working backwards

Nina spent half her money on a game, then $7 on lunch, and has $18 left. How much did she start with?

  1. Start from the end: $18 left. Working backwards means beginning with the result.
  2. Undo the lunch: 18 + 7 = 25 She had $25 before lunch.
  3. Undo the half: 25 × 2 = 50 $25 was half of what she had, so double it.
  4. Check forwards: half of 50 is 25, minus 7 is 18 ✓ Always run it forwards to confirm.

$50

A simpler case first

At a party, every pair of the 12 guests shakes hands once. How many handshakes?

  1. Try small: 2 people give 1; 3 give 3; 4 give 6; 5 give 10. Build a table you can actually count.
  2. Differences are 2, 3, 4 — each new person shakes everyone already there. The pattern has a reason, which is what makes it trustworthy.
  3. So the total is 1 + 2 + ⋯ + 11 for 12 people. The 12th person adds 11 new handshakes.
  4. 11 × 122 = 66 Or directly: each of 12 shakes 11 hands, then halve for double-counting.

66 handshakes.

Guess, check, revise

Tickets cost $8 for adults and $5 for children. 20 tickets sold for $136. How many adults?

  1. Guess 10 adults: 10(8) + 10(5) = 130. Too low by 6. Start in the middle and see which way to move.
  2. Each adult swapped in for a child adds $3. 8 - 5 = 3 — this is what makes the next guess exact.
  3. 6 ÷ 3 = 2 more adults, so 12. Use what the check told you rather than guessing blind.
  4. Check: 12(8) + 8(5) = 96 + 40 = 136 ✓ Exactly right.

12 adults.

Words to know

Strategy
A general approach — draw it, tabulate it, work backwards, try a simpler case.
Work backwards
Start from the result and undo each step.
Guess, check, revise
Make a sensible guess, test it, and use what you learn to guess better.
Simpler case
Solve the same problem with small numbers, find the pattern, then scale up.

Mistakes to watch for

Starting to calculate before deciding what is being asked. Numbers on the page invite arithmetic, and arithmetic on the wrong quantity is still wrong. Say the question in your own words first.

Working backwards but undoing in the original order. If the steps were halve, then subtract 6, the undo is add 6, then double. Reversing the order is the whole method.

Finding a pattern from two terms only. 2, 4, … could continue 6, 8 or 8, 16. Always test a candidate rule on a third term before building on it.

Skipping the check. A sensible-looking answer to a slightly different question is the most common way to lose marks, and re-reading the question takes ten seconds.

George Pólya set out those four steps in How to Solve It (1945). It has never been out of print and has sold over a million copies.

His most quoted advice is to ask, when stuck, "is there a related problem I already know how to do?"

He also recommended solving a deliberately easier version first — smaller numbers, fewer cases — which is why "try a simpler case" appears in this unit.

Browse all courses