Consider the integer lattice $Z^2$. This is a graph with edges between neighboring vertices at integer coordinates. If we remove each edge independently with probability $p=1/2$, we obtain a random subgraph. With positive probability, the left side of the square $[-n,n]^2$ will still be connected to the right side in this random subgraph, but no one knows how many edges there are in a typical connection. I will discuss what is known about this quantity, called the chemical distance, and talk about recent results with Michael Damron and Jack Hanson which give new estimates for this distance.