Competition · AMC preparation · step 4 of 4

AMC 8 · 2004 · #18

Grade 3 counting

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

logical-deductionset-partitionsystematic-enumerationcasework caseworksystematic-enumeration ↑ Prerequisites: systematic-enumerationmulti-digit-arithmetic

📏 Medium solution 💡 3 insights

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Problem

Five friends — Alice, Ben, Cindy, Dave, Ellen — each throw two darts at a target whose regions are worth 1 through 10 points. Across all 10 throws, every region is hit exactly once. Each person's score is the sum of their two regions. Alice scored 16, Ben 4, Cindy 7, Dave 11, and Ellen 17. Who hit the region worth 6?

Pick an answer.

(A)

Alice

(B)

Ben

(C)

Cindy

(D)

Dave

(E)

Ellen

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Use Matrix Logic

This is a classic "who has what?" puzzle: five people, ten numbered regions, and the constraint that each region is used exactly once. Tool #4 (Use Matrix Logic) is the natural primary — we track which numbers each person could still own and force assignments as evidence builds. To feed the matrix, Tool #2 (Make a Systematic List) writes out every distinct pair {a,b} with a+b equal to each score. Tool #3 (Eliminate Possibilities) then crosses off any pair containing a number that some earlier assignment has already locked in. Starting from the person with the fewest possible pairs (Ben, who has only one) cascades to a unique solution.

1STEP 1

List the pairs for each score

For each score, list every region pair 1–10 that sums to it — the candidate matrix.

Score & Possible pairs ; Ben 4 & {1,3} ; Cindy 7 & {1,6}, {2,5}, {3,4} ; Dave 11 & {1,10}, {2,9}, {3,8}, {4,7}, {5,6} ; Alice 16 & {6,10}, {7,9} ; Ellen 17 & {7,10}, {8,9}

2STEP 2

Lock in Ben

Ben's 4 has only the pair {1,3}, so Ben hit 1 and 3; cross out every pair using them.

Ben={1,3} → used={1,3}. Cross out {1,6} and {3,4} from Cindy; cross out {1,10} and {3,8} from Dave.

3STEP 3

Lock in Cindy

That leaves Cindy only {2,5}, so Cindy hit 2 and 5; erase pairs using them.

Cindy={2,5} → used={1,2,3,5}. Cross out {2,9} and {5,6} from Dave.

4STEP 4

Lock in Dave

Only {4,7} dodges the used numbers, so Dave hit 4 and 7; regions 6, 8, 9, 10 remain.

Dave={4,7} → used={1,2,3,4,5,7}. Remaining regions: {6,8,9,10}.

5STEP 5

Split the leftover numbers

From {6,8,9,10}, only 6+10=16 fits Alice and 8+9=17 fits Ellen.

Alice={6,10}, Ellen={8,9}

6STEP 6

Find who hit 6

The 6 lives in the pair 6 and 10, which belongs to Alice.

6 ∈ Alice's pair → (A) Alice

Answer

Alice

Verify the full assignment satisfies every condition. Ben 1+3=4 ✓, Cindy 2+5=7 ✓, Dave 4+7=11 ✓, Alice 6+10=16 ✓, Ellen 8+9=17 ✓. The regions used are {1,3,2,5,4,7,6,10,8,9}={1,2,…,10} — every region hit exactly once ✓. Also, 1+2+…+10=55 and 4+7+11+16+17=55 ✓. The other answer choices fail: Ben's pair is {1,3} (no 6), Cindy's is {2,5} (no 6), Dave's is {4,7} (no 6), Ellen's is {8,9} (no 6). Only Alice owns the region worth 6, so (A) is forced.

💡Key takeaway

When every clue is a sum and every number is used exactly once, start with the person whose score has only one possible pair — that one fact unlocks the next, and the rest falls like dominoes.

List the pairs for each score
Lock in Ben
Lock in Cindy
Lock in Dave
Split the leftover numbers
Find who hit 6

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