Problem:
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 as a digit (such as or is a multiple of that person leaves the circle and the counting continues. Who is the last one present in the circle?
Answer Choices:
A. Arn
B. Bob
C. Cyd
D. Dan
E. Eve
Solution:
Notice that the first numbers that contain a as a digit or are multiples of are , and . 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 people, starting with (Arn). After everyone says a number, must say , so he leaves the circle. The circle now has members: (Bob), (Cyd), (Dan), (Eve), and (Fon) -- with restarting his counting at . Everyone in the circle counts without incident, and it loops around such that Bob says . However, this leaves Cyd to say , and he leaves the circle. The circle now has members: -- with continuing the counting at . says , and says , and therefore leaves the circle. The circle now has members: -- with continuing the counting at . The counting loops around, and says , and therefore leaves the circle. The circle now has members: -- with starting at . They go back and forth, with saying even numbers and saying odd numbers. As such, eventually, must say , and as such, leaves the circle. This makes -- Dan -- the last one left in the circle. Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions