AMC8 2018 8, Problem 3

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?

Solution: The five numbers which cause people to leave the circle are $7, 14, 17, 21,$ and $27.$ The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count. Assuming the six people start with $1$, Arn counts $7$ so he leaves first. Then, Cyd counts $14$ as there are $7$ numbers to be counted from this point. Then, Fon, Bob, and Eve, count, $17,$ $21,$ and $27$, respectively, so the last one standing is Dan. Hence, the answer would be $\fbox{\text{Dan}}$.

Answer: \text{Dan}