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Competition maths
Competition problems are rarely about knowing a special formula. They are about having a few good habits for getting unstuck. Here is a starter toolkit.
The problem-solving toolkit
- Understand the problemSay it in your own words. What is given, and what exactly are you asked to find?
- Draw a picture or make a tableA diagram or an organised list often shows the structure that words hide.
- Try small casesSolve the problem for 1, 2, 3 and 4 before trying 100. Then look for a pattern.
- Work backwardsStart from the answer you want and ask what would have to be true just before it.
- Guess, check and improveA sensible first guess, tested honestly, tells you which way to adjust.
- Look for symmetryIf two parts of the problem behave the same way, you may only need to solve one.
- Check your answerDoes it make sense? Does it work in the original problem, not just in your equation?
Worked example: the handshakes
Ten people meet and each person shakes hands once with every other person. How many handshakes are there in total?
- Try small cases. With 2 people there is 1 handshake. With 3 people there are 3. With 4 there are 6. With 5 there are 10.
- Look for the pattern. Each new person shakes hands with everyone already there, so the totals grow by 1, then 2, then 3, then 4.
- Find the rule. With n people each person shakes hands with n − 1 others, but that counts every handshake twice, so the total is n × (n − 1) ÷ 2.
- Answer. For 10 people that is 10 × 9 ÷ 2 = 45 handshakes.
Competitions and problems to try
- CEMC Problem of the Week (University of Waterloo) (opens in a new tab) · A new problem each week for grades 3 to 12, with solutions the week after
- CEMC maths contests (University of Waterloo) (opens in a new tab) · Past contests and practice papers
- ISMTF competitions (opens in a new tab) · International schools maths competitions, including those held in Switzerland
- Crux Mathematicorum (opens in a new tab) · Free problem-solving journal from the Canadian Mathematical Society, with an Olympiad Corner
- UKMT (opens in a new tab) · UK Maths Trust challenges and past papers
- NRICH (opens in a new tab) · Rich problems and games from Cambridge