Problem 3 · 2018 AMC 8
Medium
Logic & Word Problems
caseworkcareful-counting
Students Arn, Bob, Cyd, Dan, Eve, and Fon are arranged in that order in a circle. They start counting: Arn first, then Bob, and so forth. When the number contains a 7 as a digit (such as 47) or is a multiple of 7 that person leaves the circle and the counting continues. Who is the last one present in the circle?
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Answer: D — Dan.
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Hint 1 of 2
The work isn't in the counting — it's in writing the "danger" numbers first: 7, 14, 17, 21, 27… Once you have that short list, you only care about who says those numbers, not the boring ones in between.
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Hint 2 of 2
The technique is a simulation: keep a shrinking list of who's still in, and remember the count keeps climbing while the people loop — restart from whoever is next, never from 1.
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Approach: step through the counting, eliminating people
- Step 1 is the setup that makes everything easy: list the eliminating counts in order — 7, 14, 17, 21, 27, … (multiples of 7, or any number containing the digit 7). Now you only have to find who says each of these.
- Pass 1 (6 people): A=1, B=2, C=3, D=4, E=5, F=6, A=7 ⇒ A out.
- Pass 2 (5 people: B, C, D, E, F): B=8, C=9, D=10, E=11, F=12, B=13, C=14 ⇒ C out.
- Pass 3 (4: B, D, E, F): D=15, E=16, F=17 ⇒ F out.
- Pass 4 (3: B, D, E): B=18, D=19, E=20, B=21 ⇒ B out.
- Pass 5 (2: D, E): D=22, E=23, D=24, E=25, D=26, E=27 ⇒ E out.
- Last remaining: Dan.
- You'll see it again: "person leaves the circle, counting continues" problems are simulations — resist hunting for a formula and just bookkeep carefully; the only trap is forgetting the count keeps rising as people drop out.
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