Here are some diagrams that represent the possible paths of length 2n from one corner of an n-by-n grid to the opposite corner. The number of paths are the central binomial coefficients
central meaning they fall along the center line of Pascal's triangle.
(Oddly enough, the Catalan numbers describe how many of these paths stay under the main diagonal.)
1 x 1 grid, 2 paths:
2 x 2 grid, 6 paths:
3 x 3 grid, 20 paths:
4 x 4 grid, 70 paths:
5 x 5 grid, 252 paths:
Designed and rendered using Mathematica 3.0 for the Apple Macintosh.
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