Pigeonhole and worst case — Maths Olympiad practice
“How many must you take to be sure?” is a favourite Olympiad puzzle. Imagine you are as unlucky as possible: count how many you could take and still not have what you want. One more than that is always enough.
✓ You passed this topic’s test.
How to solve it
Think of the unluckiest case — “take everything that does not help, then one more”
Imagine the unluckiest draws: how many can you take and still NOT have what you want?
Add 1 to that number — the next one must give it.
Pigeonhole: with n boxes (colours, months, days), n × (k − 1) + 1 things make sure some box has at least k.
Use it for: Socks, gloves, coloured balls, birthdays and “to be sure” or “at least” questions in Class 5–10 reasoning papers.
Worked example: A drawer has lots of red, blue and green socks, all mixed up. In the dark, how many socks must you take out to be sure of getting a matching pair? → 4
Worst case: the first 3 socks are one red, one blue and one green — no pair yet.
There are only 3 colours, so the 4th sock must match one of them.
Check: With 3 socks you might still have no pair, so 3 is not enough; 4 always works ✓
✏️ Practice
Work it out in your head or on paper, type the answer and press Check. Stuck? Press “Show me how”.
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🏁 Topic 23 test
20 questions. Check each answer, then go to the next one. Get 16 or more right to pass.
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🧦 Sock Sorter badge when you pass — and a certificate when you finish all 13 topics of
Part 4 · Class 5–10 problem solving. You can still take the test without signing in.
🖨️ Pigeonhole Principle Worksheet for Maths Olympiad
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