A drawer holds 10 red socks, 10 blue socks and 10 green socks, all mixed up in the dark. How many socks must you take out to be sure of having a matching pair?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Imagine the unluckiest possible draw.
2The strategy
Ask how many socks you could take while still having no pair at all. One more than that number forces a match, because there is nowhere new for it to go.
3The full solution
The worst case is one sock of each colour, which is 3 socks and still no pair. The fourth sock must repeat a colour, so 4 socks are enough.
Solution
The reliable way
Think about the worst case rather than the lucky one. You could draw one red, one blue and one green - three socks with no matching pair. But a fourth sock must be red, blue or green, so it repeats a colour you already hold. Four socks therefore guarantee a pair, and three do not. The transferable idea: fill every box once, then add one.
The elegant way
With 3 colours acting as boxes, 4 socks cannot all go in different boxes, so two share a box - that is the pigeonhole principle in one line.
Why this is on the test: To be sure means the worst case, and answering with the lucky case is the single most common misreading in the counting strand.