Problem 2 · AMC 8 Stretch
Core
Counting & Probability
Logic & Word Problems
considering-extreme-casesaccounting-for-all-possibilitieslogical-reasoning
A drawer has 7 blue socks and 7 red socks, all jumbled together. You reach in (in the dark) and pull out socks. (1) How many socks must you grab to be CERTAIN of getting a matching pair of some color? (2) Now a harder, different question: how many must you grab to be CERTAIN of getting two BLUE socks specifically? (Imagine the worst possible luck.)
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Answer: Any matching pair: 3 socks. Two blue socks specifically: 9 socks
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Hint 1 of 4
'A matching pair of some color' means two blues OR two reds. There are only two colors. Think about the worst case: what is the most socks you could grab and still NOT have a pair?
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Hint 2 of 4
If you grabbed 2 socks of different colors (one blue, one red), you have no pair yet. But the very next sock must match one of them!
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Hint 3 of 4
For part (2), 'two blue' is much pickier. Worst luck: you keep pulling out red socks. How many reds are in the drawer? You might pull every one of them before a blue shows up.
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Approach: Considering the worst case — pigeonhole vs. a specific color
- Part 1, a matching pair of any color: with only two colors, after you grab 2 socks the unluckiest result is one blue and one red — no pair yet. But the 3rd sock has to be blue or red, so it MUST match one of the two you already hold. So 3 socks guarantee a matching pair. (You can also list the patterns of 3 socks: BBB, BBR, BRR, RRR — every one contains a pair.)
- Part 2, two BLUE socks specifically: this is a pickier demand. Imagine pulling out reds again and again with terrible luck. There are 7 red socks, so you could pull all 7 reds before any blue appears. After those 7 reds you still need 2 blue socks, so in the worst case you need \(7+2=9\) socks.
- The lesson: read the question carefully! 'A matching pair of any color' (3 socks) and 'two of a specific color' (9 socks) have completely different answers.
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