A hard question on pigeonhole principle for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 qualifying rounds and finals. The trick is to imagine the unluckiest case.
The question
A lucky-draw box holds 20 cards numbered 1 to 20, one of each.
Fan draws them one at a time without looking.
What is the fewest she must draw to be sure two of her cards add up to 12?
Show the hints (3)
💡Hint 1: Which numbers can pair with another to make 12? Which can never pair?
💡Hint 2: Group the numbers that can pair, like 1 with 12 minus 1. What about 6? It would need itself, and there's only one. And numbers from 12 up?
💡Hint 3: Worst case: one card from every pair, plus every card that can't pair: 15 in all.
Worked solution
Have a go first, then read the solution step by step.
1.There are 5 pairs making 12: 1 and 11, 2 and 10, …, 5 and 7.
2.Numbers that can't pair: 6 (only one card); 12 to 20, 9 of them: 10 in all.
3.Worst luck: one from each pair and all the loners, 5 + 10 = 15, still no two making 12.
4.The next card must complete a pair: 15 + 1 = 16.
⚠️Common mistake: thinking only of the pairs, forgetting that numbers from 12 up and 6 never pair and can all be drawn.
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