Problem 21 · 2011 AMC 8
Medium
Logic & Word Problems
constraint-satisfaction
Students guess that Norb's age is 24, 28, 30, 32, 36, 38, 41, 44, 47, and 49. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
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Answer: C — 37.
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Hint 1 of 2
Turn each sentence into a numeric filter, then apply them one at a time — the first clue alone slices the list nearly in half.
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Hint 2 of 2
‘At least half of 10 guessed too low’ means at least 5 guesses are below his age, so his age beats the 5th-smallest guess. ‘Two are off by one’ means his age has a guess just below and just above it — it's wedged inside a pair of guesses differing by 2.
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Approach: translate each clue into a constraint and filter in order
- Sort: 24, 28, 30, 32, 36, 38, 41, 44, 47, 49. ‘At least half too low’ needs ≥5 guesses below his age, so age > the 5th value, 36.
- ‘Two off by one’ means a guess sits 1 below and 1 above his age — so his age is the middle of a pair 2 apart. Above 36 the only such pairs are (36, 38) giving 37, and (47, 49) giving 48.
- ‘Prime’ decides it: 37 is prime, 48 is not.
- Norb is 37.
- Worth keeping: logic puzzles are just translation — rewrite every clue as ‘age > 36,’ ‘age between two close guesses,’ ‘age prime,’ and intersect the survivors.
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