Problem 18 · 2004 AMC 8
Hard
Logic & Word Problems
logic-puzzleelimination
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a different value. The scores are: Alice 16, Ben 4, Cindy 7, Dave 11, Ellen 17. Who hits the region worth 6 points?
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Answer: A — Alice.
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Hint 1 of 2
Every region 1–10 is hit exactly once across all ten darts — so once a number is 'claimed', it's gone for everyone else. Attack the score with the fewest possible pairs first: that's the smallest total, Ben = 4.
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Hint 2 of 2
The technique is most-constrained-first (like solving a Sudoku): handle the score that has only one possible pair, lock those numbers away, and the next score usually collapses to a single option too — a chain reaction.
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Approach: most-constrained-first chain
- Ben = 4 with two different regions: the only option is 1 + 3 (2 + 2 isn't allowed). Lock away 1 and 3.
- Cindy = 7 could be 1+6, 2+5, or 3+4 — but 1 and 3 are now gone, leaving only 2 + 5. Lock away 2 and 5.
- Dave = 11 from what's left {4,6,7,8,9,10}: only 4 + 7 survives (5+6 and 2+9 use claimed numbers). Lock away 4 and 7.
- Alice and Ellen must split the remaining {6, 8, 9, 10}. Alice = 16 forces 6 + 10 (since 7+9 uses a gone 7), and Ellen = 17 = 8 + 9.
- So Alice is the one who hits 6.
- Why start small: the smallest and largest totals have the fewest valid pairs, so they're the safest place to begin — each forced pair removes numbers and tightens the next one. You'll use this 'pin the most-restricted thing first' habit in every logic-grid puzzle.
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