AMC 10 · 2013 · #6

Grade 1 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logical-deductionparity casework ↑ Prerequisites: logical-deduction

📏 Medium solution 💡 2 insights

Problem

Six brothers, one of them Joey, have ages 3, 5, 7, 9, 11, and 13. Two of them whose ages add to 16 went to the movies. Two more, each younger than 10, went to play baseball. Joey and the 5-year-old stayed home. Find Joey's age.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

The set of ages is small and fixed, and the answer is one of five choices — the classic trigger for Tool #3 (Eliminate Possibilities). First Tool #2 (Make a Systematic List) pins down every pair that could be the movie pair by adding to 16; then Tool #3 knocks out the ones that clash with the other clues until only one arrangement survives. Joey is whoever is left standing.

1STEP 1

List every pair that sums to 16

The movie pair sums to 16, and only three pairs of ages do: 3+13, 5+11, and 7+9.

3+13=16, 5+11=16, 7+9=16

2STEP 2

Cross off the pair using the 5-year-old

The 5-year-old stayed home, so 5+11 is out and the movie pair is 3+13 or 7+9.

home = {5, Joey} → 5 ∉ movies → movies ∈ {(3,13), (7,9)}

3STEP 3

Test which movie pair leaves a legal baseball pair

Baseball takes two of {3, 7, 9}; if the movies took 7+9, only 3 is left, so the movie pair must be 3+13.

under 10 = {3,7,9}; movies=(7,9)→ baseball⊆{3} (too few) → movies=(3,13)

4STEP 4

Place the rest; Joey is who remains

Movies take 3 and 13, baseball takes 7 and 9, home has the 5-year-old and Joey, so Joey's age is 11.

movies={3,13}, baseball={7,9}, home={5, Joey}→ Joey=11 (D)

Answer

Check every clue against the final arrangement: movies = 3+13 = 16 (sum is 16, good); baseball = 7 and 9, both under 10 (good); home = the 5-year-old and Joey (11). All six brothers 3,5,7,9,11,13 are used exactly once, and each clue is satisfied, so Joey being 11 is consistent.

💡Key takeaway

Write down every pair that fits a clue, cross off the ones the other clues forbid, and the brother with no group left is your answer.

List every pair that sums to 16
Cross off the pair using the 5-year-old
Test which movie pair leaves a legal baseball pair
Place the rest; Joey is who remains

List every pair that sums to 16

Cross off the pair using the 5-year-old

Test which movie pair leaves a legal baseball pair

Place the rest; Joey is who remains

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