Problem:
Alice and Bob play a game involving a circle whose circumference is divided by equally-spaced points. The points are numbered clockwise, from to . Both start on point . Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves points clockwise and Bob moves points counterclockwise. The game ends when they stop on the same point. How many turns will this take?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Write the points where Alice and Bob will stop after each move and compare points.
So Alice and Bob will be together again after six moves.
If Bob does not move and Alice moves points or points each time, they will still be in the same relative position from each other after each turn. If Bob does not move, they will be on the same point when Alice first stops on point , where she started. So Alice will have to move steps times to stop at her starting point.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions