Problem 12 · AMC 8 Stretch
Core
Counting & Probability
asking-key-questionspigeonhole
An apartment building has \(20\) mailboxes. How many letters must the mailman deliver to be CERTAIN that at least one box ends up with \(3\) or more letters?
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Answer: 41 letters
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Hint 1 of 3
Ask the key question: what is the most letters you could deliver while keeping every box at \(2\) or fewer (so no box has reached \(3\) yet)?
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Hint 2 of 3
The worst case puts exactly \(2\) letters in every single box. With \(20\) boxes, how many letters is that in total?
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Hint 3 of 3
One more letter after that must push some box up to \(3\).
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Approach: Asking the key question — the pigeonhole worst case
- Ask the key question: how many letters can be delivered without any box reaching \(3\)?
- In the worst case, every one of the \(20\) boxes gets exactly \(2\) letters: \(20 \times 2 = 40\) letters, and still no box has \(3\).
- But the very next letter, the \(41\)st, has to go into a box that already holds \(2\), making it \(3\). So \(41\) letters guarantee that some box has at least \(3\).
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