Problem 8 · 2024 AMC 8
Medium
Counting & Probability
careful-countingcasework
On Monday Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
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Answer: D — 6 different amounts.
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Hint 1 of 2
Each day has 2 choices over 3 days, so 2×2×2 = 8 paths — but the question asks for AMOUNTS, not paths. Why might those two counts differ?
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Hint 2 of 2
Technique: track the SET of reachable amounts day by day, not the branching paths. When two paths land on the same dollar amount, the set automatically merges them — so you can't over-count.
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Approach: track the set of reachable amounts, letting duplicates merge
- The trap is counting 2×2×2 = 8 paths; the question wants distinct amounts, and some paths collide. So carry a SET forward each day instead of a list of paths. Start $2. Tuesday: 2+3 = 5 or 2×2 = 4 → {4, 5}.
- Wednesday, apply both moves to each: 4+3 = 7, 4×2 = 8, 5+3 = 8, 5×2 = 10. The two 8's merge → {7, 8, 10}.
- Thursday from {7, 8, 10}: 7→{10,14}, 8→{11,16}, 10→{13,20}, all distinct → {10, 11, 13, 14, 16, 20}.
- 6 distinct amounts. This transfers: whenever "how many outcomes" can repeat, count states (the set), not branches — the set throws away duplicates for you. It's the seed of breadth-first thinking.
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