What is the expected number of turns for two dots to meet

Question

Let's say I have a circle with $r$ dots $0,...,r-1$. I drop pin $1$ at a random dot $i$ on the circle and pin $2$ at a random dot $j$ in the circle. Now I rotate pin $1$ clockwise moving it one step clockwise at a turn and I rotate pin $2$ clockwise $2$ pints ever turn. How can I calculate the expected number of turns I do until both pins meet for the very first time?

Answer 1

Moving pin $1$ clockwise at a rate of $1$, and pin $2$ clockwise at a rate of $2$ is the same as keeping pin $1$ still, and moving pin $2$ clockwise at a rate of $2-1=1$. Does that help? Do you know how to do the expected value calculation from there? Just think of pin $1$ as fixed, and think about the number of steps to reach it from all of the different places pin $2$ could land.