Path Counting - Chances of two people meeting?

[phono]

Homework Statement

I am having trouble with this problem.

A network of city streets forms square bloacks as shown in the diagram below.
http://img182.imageshack.us/my.php?image=librarypoolqs6.jpg

Jeanine leaves the library and walks toward the pool at the same time as Miguel leaves the pools and walks toward the lbrary. Neither person follows a particular route, except that both are always moving toward their destination. What is the probability that they will meet if they both walk at the same rate?

In addition, how would I solve this for a 1 by 1 grid, 2 by 2 grid, 3 by 3 grid,etc.?

I know that you have to use Pascal's Triangle and I think that they would have to meet on their "4th" moves. The answer in the book is 35/128 but I don't know how to get this.

 

[Dick]

First figure out which corners can be reached in 4 moves. Then figure out the number of ways to reach each of those corners (this is where Pascal's triangle comes in). Now figure out the probability that each walker will land at a given corner and add them up.