Meeting in the middle
Charlie and Gabriel are walking towards each other's houses. What is the chance that they'll meet along the way?
Problem
Charlie and Gabriel live in opposite corners of a town. The town is built on a grid system, and the roads look like this:
Charlie and Gabriel leave their houses at the same time. Each begins to walk towards the other's house by walking along the sections of road, moving at the same speed. They always take the shortest route possible (and at every crossroads they make their next choice randomly, if there are multiple shortest routes).
What is the probability that they will meet at some point in the journey?
How does the probability change if the size of the grid increases to three squares by three squares? What about four squares by four squares?
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