Dr. Christopher Bird

The Random Maze Problem - Variations

My original posting of the Random Maze Problem has been popular with my readers and viewers, and has apparently been the subject of much discussion among amateur mathematicians. As yet no one has provided a solution for it, but many are continuing in their efforts. 

However since it brought such joy and entertainment, I thought I would post a few interesting modifications and variations that people have been suggesting. I suspect that these are all more difficult than the original problem, but they are worth attacking as well.

1. One Way Maze: I quite enjoy this variation. Instead of taking p as the probability that a unit line is open and can be traversed, suppose that there are four different probabilities. We take p1 as the probability that a vertical line can be traversed left to right, and p2 the probability that it can be traversed right to left. In the same way, p3 and p4 represent the probability that a horizontal line can be traversed up or down. In this more complicated system, what is the probability of solving the maze?
2. The Generic Maze: This is a more general version of the original random maze, and simply asks what is the probability of being able to move from a specific point (x,y) in the maze to another point (w,z). The original problem uses the points (1,1) and (N,M), but it is equally interesting to look at other points.
3. Exit Maze: The original problem looked at solutions from one corner to the other diagonally opposite corner. But it is also interesting to start in the center of the maze, and calculate the probability that there exists a path to the boundary.