Problem 11 · AMC 8 Stretch
Core
Counting & Probability
Logic & Word Problems
asking-key-questionspigeonholeconsidering-extreme-cases
A drawer has \(7\) pairs of blue socks and \(7\) pairs of red socks, all jumbled together. Reaching in the dark, how many socks must you grab at once to be SURE you have a matching pair (two of the same color)?
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Answer: 3 socks
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Hint 1 of 3
The numbers \(7\) and \(7\) are a distraction. Ask the key question: assuming the worst possible luck, how many socks could you pull out and STILL not have a matching pair?
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Hint 2 of 3
There are only two colors. The most you could grab without a match is one blue and one red — just \(2\) socks.
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Hint 3 of 3
Now think about the very next sock. What color can it possibly be?
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Approach: Asking the key question — the pigeonhole worst case
- Ask the key question: with the worst luck, how many socks can you draw and still have no match?
- There are only two colors. The unluckiest grab gives you one blue and one red — \(2\) socks, no match yet.
- But the third sock you pull MUST be blue or red, and either way it matches one you already hold. So \(3\) socks are always enough (and \(2\) is not). The numbers \(7\) and \(7\) never matter — only that there are \(2\) colors.
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