Earlier this week I wrote an article about an interesting (and possibly previously unpublished) about a mathematics puzzle that so far as I know does not have a simply solution. Apparently it was popular, and as such I have decided to devote today's article to another interesting puzzle. As with the previous one, I have asked a number of mathematician friends about this one and as yet I have not found a previously published source, or a solution for it. Perhaps one of you reading this will find the answer...

Now take your favorite random boolean number generator (ie a coin). For each two neighbouring points, either horizontal or vertical, flip the coin. If it comes up heads, leave the space blank and if it comes up tails draw a line between those two points. In the end, you should have something like this:

The question is, what is the probability that the final figure will be a solvable maze? By solvable here, I mean start in the upper left corner, and draw a path through the figure to the lower right corner without passing through any of the lines. The particular example above is not solvable, whereas the one below is:

The full puzzle though is slightly more complicated. For arbitrary N and M, and for arbitrary probability p of drawing any given line (the example above uses p = 0.5), what is the probability that the final maze has a solution?

This is an interesting question, because it is simple to state and yet so far the mathematicians I have shown it to have not been able to solve it. In fairness, I don't have a really good, simple solution either (It can be solved with more complicated computer algorithms).

So I leave it to the readers of this article to see if they can find a solution. I will be interested to see where this puzzle leads...