### A Challenge for my Readers!

In this blog post, I present a very interesting math problem that I came up with while pondering random walk problems.

You are walking along the x axis and you start at $x=0$. At each time step you move to $x+1$ with probability $ \frac{1}{1+x} $ and to $ x-1 $ with probability $ \frac{x}{x+1} $. What is the expected number of steps until you return to $ 0 $ for the first time? What makes this problem harder than a typical random walk problem is that the probability of moving to $x+1$ vs $x-1$ is not fixed -- it changes based on your position. That being said, the typical counting methods for solving problems of this type do not apply. There is a clever and fascinating solution that doesn't require too much advanced knowledge, but I will leave it to you to think about this problem.

You are walking along the x axis and you start at $x=0$. At each time step you move to $x+1$ with probability $ \frac{1}{1+x} $ and to $ x-1 $ with probability $ \frac{x}{x+1} $. What is the expected number of steps until you return to $ 0 $ for the first time? What makes this problem harder than a typical random walk problem is that the probability of moving to $x+1$ vs $x-1$ is not fixed -- it changes based on your position. That being said, the typical counting methods for solving problems of this type do not apply. There is a clever and fascinating solution that doesn't require too much advanced knowledge, but I will leave it to you to think about this problem.