The way to get a closed form solution for the expected time is to note that there are 2N + 1 states. N states for each notch and each direction, and one "final"

The way to get a closed form solution for the expected time is to note that there are 2N + 1 states. N states for each notch and each direction, and one "final" absorbing state for the end of the maze. You can write a matrix A where the transition probability between states i and j is given by Aij. Then A^n gives the distribution over states after n steps. It's been a while since I've done this, but I think the solution is something like the last index in the vector E[steps] = (I - A)^-1 * (1, 1, 1, 1 ...) I-A is called the "fundamental matrix" of the markov chain described by Aij. EDIT I wrote some numpy to verify that this solution matches the empirical growth rate of 1.4 found in the video: https://gist.github.com/mgraczyk/7917a3322f65fe60fa333da7936f4cee https://gist.github.com/mgraczyk/7917a3322f65fe60fa333da7936...