Problem 6 · 2020 AMC 8
Medium
Logic & Word Problems
casework
Aaron, Darren, Karen, Maren, and Sharon rode on a small train that has five cars that seat one person each. Maren sat in the last car. Aaron sat directly behind Sharon. Darren sat in one of the cars in front of Aaron. At least one person sat between Karen and Darren. Who sat in the middle car?
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Answer: A — Aaron.
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Hint 1 of 2
Lock down the rigid clue first: “Aaron directly behind Sharon” glues them into a single block (Sharon then Aaron). A block of 2 has very few places it can sit, so that's your lever.
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Hint 2 of 2
Slide the Sharon–Aaron block across the cars and keep only the spot that still leaves room for Darren ahead of Aaron and Karen at least one car from Darren.
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Approach: glue the rigid pair into a block, then slide it
- Start with the strictest clue: “directly behind” glues Sharon–Aaron into one block (S then A). Maren is fixed in car 5. A glued block of 2 only fits a handful of ways — test each.
- S, A in cars 1, 2: Darren needs a car ahead of car 2, but car 1 is taken. Fail.
- S, A in cars 3, 4: Darren and Karen must fill cars 1 and 2 — adjacent, with nobody between them. Breaks the spacing rule. Fail.
- S, A in cars 2, 3: Darren in car 1, Karen in car 4 — three cars apart, fine. The row is Darren, Sharon, Aaron, Karen, Maren.
- Only that arrangement survives, so Aaron is in the middle car.
- You'll see this again as: in seating/ordering logic, attack the most restrictive clue first (a fixed seat, or two people who must be adjacent). It chops the possibilities fastest, so you test a few cases instead of all 120 orderings.
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