Question

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?

$\text{Arn}$

$\text{Bob}$

$\text{Cyd}$

$\text{Dan}$

$\text{Eve}$

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Written solution:

Notice that the first 5 numbers that contains a $7$ as its digit or are a multiple of $7$ are $7,14,17,21,27.$ Any player who lands on one of these numbers must leave the circle.

With this in mind, let's start counting. Initially, we have all $6$ people, starting with $A$ (Arn). After everyone says a number, $A$ must say $7,$ so he leaves the circle.

The circle now has $5$ members: $B$ (Bob), $C$ (Cyd), $D$ (Dan), $E$ (Eve), and $F$ (Fon) -- with $B$ restarting his counting at $8.$ Everyone in the circle counts without incident, and it loops around such that Bob says $13.$ However, this leaves Cyd to say $14,$ and he leaves the circle.

The circle now has $4$ members: $B,D,E,F$ -- with $D$ continuing the counting at $15.$ $E$ says $16,$ and $F$ says $17,$ and therefore leaves the circle.

The circle now has $3$ members: $B,D,E$ -- with $B$ continuing the counting at $18.$ The counting loops around, and $B$ says $21,$ and therefore leaves the circle.

The circle now has $2$ members: $D,E$ -- with $D$ starting at $22.$ They go back and forth, with $D$ saying even numbers and $E$ saying odd numbers. As such, eventually, $E$ must say $27,$ and as such, leaves the circle. This makes $D$ -- Dan -- the last one left in the circle.

Thus, D is the correct answer.

2018 AMC 8 Problem 3

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Problem 3 in Other Years