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So is your conjecture that in the hex-grid (or triangular grid depending on
how you look at it), there are no two right triangles that share a
hypotenuse with odd squared length?

EDIT: I should probably add "where the legs of the triangles are along the orthogonal and diagonal directions".

More edit: and to avoid the trivial swapping of the order of diagonal/orthogonal steps, make it "nonisomorphic triangles"