Start with a square grid cut from graph paper, say N x N. Mark the lower left intersection as the origin, so that the x and y coordinates each go from 0 to N. From the origin to any other intersection in the square there are a number of ways of traversing the grid lines to get there in the shortest possible path. For example, there are six ways of getting to (1,5) from the origin, and there are also six ways of getting to (5,1) and likewise for (2,2). So there are three intersections where the number of ways is six.
For a particular value of N, there are four intersections with the same number of ways to get there via a shortest path such that these intersections all occur within the top-right quarter of the square, or at least on its border. That is, all four have x≥N/2, y≥N/2. The number of ways in question is less than 10,000.
What is the number of different shortest routes to each of these intersections?