A medium question on pigeonhole principle for P4–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 teacher hands out all 17 books to a class, any number to each pupil, even none.
She says, "However they're shared, some pupil gets at least 3."
She is right.
What is the largest the class could be?
Show the hints (2)
💡Hint 1: If everyone gets fewer than 3 — at most one less than 3 — how many books can the class take at most?
💡Hint 2: She is right, so "everyone gets fewer than 3" must be impossible: (3 − 1) × the class size must be less than 17.
Worked solution
Have a go first, then read the solution step by step.
1.With s pupils each getting at most 2, at most 2 × s books get handed out.
2.For that to fall short, 2 × s must be less than 17.
3.8 pupils: 8 × 2 = 16 is less than 17; the 1 left must go to someone, who then has 3 or more.
4.9 pupils: 9 × 2 = 18 exceeds 17, so 2 or fewer each could use them all up, nobody getting 3.
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