Answer: $2^{n-1}$ Note that Josh must pass through the center square of each row. There are 2 ways to get from the center square of row $k$ to the center square of row $k+1$. So there are $2^{n-1}$ ways to get to the center square of row $n$.
2. Josh takes a walk on a rectangular grid of $n$ rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row?