Problem 5 · AMC 8 Stretch
Core
Arithmetic & Operations
Logic & Word Problems
pattern-recognitionconsider-extreme-cases
A truck carries 4,000 crates. At its first stop it drops off half the crates. At the second stop it drops off half of what is left. At the third stop, half of the new remainder. If this pattern keeps going, at which stop will the LAST crate be dropped off?
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Answer: There is no last stop — half always remains, so the question has no answer (a peek at infinity)
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Hint 1 of 4
Don't track how many get dropped off. Track how many are still ON the truck after each stop.
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Hint 2 of 4
Start halving: 4000, then 2000, then 1000, ... What stays on the truck each time?
Still stuck? Show hint 3 →
Hint 3 of 4
After every single stop, exactly half of the crates remain on the truck. Ask yourself: can the truck ever hit zero this way?
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Approach: Track what stays, and realize it never reaches zero
- Follow the crates left on the truck. Dropping off half means half stay: \(4000\to2000\to1000\to500\to250\to125\to\dots\)
- After each stop, half of whatever is on the truck stays on the truck, so no matter how many stops happen, something is always still aboard.
- (Once the number gets odd, like 125, you can't literally split it evenly — the problem ignores that on purpose, and arguing about it is part of the fun.)
- So there is no last stop: the amount on the truck gets closer and closer to 0 but never reaches it. The 'success' here is realizing there is no answer — a neat doorway to thinking about infinity.
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