AMC 8 · 2011 · #21

Grade 4 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint

prime-numberslogical-deductionsystematic-enumeration logical-deductioncaseworksystematic-enumeration ↑ Prerequisites: prime-numbers

📏 Medium solution 💡 3 insights

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Problem

Ten students guess Norb's age: 24, 28, 30, 32, 36, 38, 41, 44, 47, 49. Norb tells them three facts about the right answer: (1) at least half of the guesses are too low, (2) exactly two guesses are off by one (one above and one below his age), and (3) his age is a prime number. Find Norb's age.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Three clean conditions slice through a finite list — perfect setup for Tool #3 (Eliminate Possibilities). Tool #2 (Systematic List) builds the short candidate set from the "off by one" clue: scan the guess list for pairs that differ by exactly 2, and the age sits squarely in the middle of each such pair. Then the "at least half too low" condition kills the candidates that are too small, and the "prime" condition finishes off whatever remains. No algebra, just three filters.

1STEP 1

Half of 10 is 5, so the five smallest guesses (24, 28, 30, 32, 36) must all be too low — Norb's age is greater than 36.

\frac{1}{2}

× 10 = 5 → age ＞ 36

2STEP 2

Both x-1 and x+1 must be guesses, so scan for pairs differing by 2: (28,30), (30,32), (36,38), (47,49) give candidates 29, 31, 37, 48.

{(28,30), (30,32), (36,38), (47,49)} → x ∈ {29, 31, 37, 48}

3STEP 3

Apply the 'more than 36' filter to the candidates: 29 and 31 drop out, leaving 37 and 48.

{29, 31, 37, 48} ∩ {x : x ＞ 36} = {37, 48}

4STEP 4

37 is prime, but 48 = 2 × 24 is composite, so only 37 survives the prime clue.

37 prime, 48 = 2 × 24 composite

5STEP 5

Norb is 37 years old, which is choice (C).

age = 37 → (C)

Answer

Verify all three clues against 37. (1) Too-low guesses: 24, 28, 30, 32, 36 — that's 5 out of 10, satisfying "at least half." (2) Off by one: 37 - 1 = 36 and 37 + 1 = 38 are both in the list — exactly two guesses off by one. (3) 37 is prime. All three clues check out, and none of the other answer choices does — for instance, 43 would need 42 and 44 both in the list, but 42 isn't there; 29 and 31 both fail the "more than 36" condition.

💡Key takeaway

This AMC 8 problem only needs Grade 4 prime-or-composite checking and a couple of "is it bigger than 36?" comparisons you already know!

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