GeekDad Puzzle of the Week – Wacky Right Integer String Triangles

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Max, making a curve out of only straight lines.
Max, making a curve out of only straight lines.

This weekend, Max and I were playing at making curves out of all straight lines. After showing him a simple Bézier curve on paper, we decided to do it with string. We created a grid of nails on a plank of wood, and the matrix included a total of 335 nails. (The grid started out as 21 x 16, but the lower right nail kept falling out, so we removed it.) It looked something a little like this:
emptygrid
Max took out some string and after a few moments, made a 3-4-5 right triangle in the corner. He noted that the length of string it took was a whole number (12), and stated that that was the only right triangle he could make with a whole number length of string.
345grid
I quickly made a second 3-4-5 right triangle elsewhere in the grid, and at that moment, this week’s GeekDad Puzzle of the Week dawned on me: Just how many distinct right triangles (of any size, shape, or orientation) could we make on this (almost) rectangular grid with integer lengths of string?

For purposes of this puzzle, assume that the nails are simply points (no width) and that no straight lines between nails needed are diverted by intermediate / uninvolved nails.

Please submit your responses to GeekDad Central by end of day Friday. As always, this week’s prize is a generously donated $50 ThinkGeek Gift Certificate, and the winner will be drawn at random from all reasonably correct responses. Good luck!

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