jmc

Question

Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise. In a turn of the game, Alice moves 5 points clockwise and Bob moves 9 points counterclockwise. The game ends when they stop on the same point. How many turns will this take?

Accepted Answer

Alice moves 5k steps and Bob moves 9k steps, where k is the turn they are on. Alice and Bob coincide when the number of steps they move collectively, 14k, is a multiple of 12. Since this number must be a multiple of 12, as stated in the previous sentence, 14 has a factor 2, k must have a factor of 6. The smallest number of turns that is a multiple of 6 is 6.