Cards numbered 1 to 10 lie face up. You take any six of them. Must two of your cards add up to 11?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Which pairs add to 11?
2The strategy
List the pairs adding to 11 and count them. Every card sits in exactly one pair, so treat the pairs as boxes and ask how many cards avoid filling a box twice.
3The full solution
The pairs are 1 and 10, 2 and 9, 3 and 8, 4 and 7, 5 and 6 - five pairs. Taking at most one from each gives only five cards.
Solution
The reliable way
The pairs adding to 11 are (1, 10), (2, 9), (3, 8), (4, 7) and (5, 6). Every card from 1 to 10 sits in exactly one of these five pairs. Suppose you took six cards and no two added to 11: then you took at most one card from each pair, which is at most five cards, fewer than six. That is impossible, so two of your six must add to 11. The transferable idea: assume what you doubt, count what it would force, and let the contradiction finish the argument.
The elegant way
Five pairs act as five boxes, and six cards cannot go into five boxes one apiece.
Why this is on the test: Proving something always happens needs an argument rather than examples, and this is the smallest setting where trying cases genuinely cannot work.