General Diagonals in Squares: Lower Secondary Mathematics Competition Question

This is the general case of the question posted earlier.

A NxN grid is drawn so that it has N^2 unit squares where N is an even number. A diagonal within a unit square is a straight line with endpoints at opposite corners of the square - one example is drawn below. You are required to draw such diagonals in such a way that none of them have a point in common; that is, two diagonals cannot overlap and they cannot touch at an endpoint.

What is the maximum number of such diagonals of unit squares that can be drawn within a general NxN grid where N is even?

What is the formula if N is an odd number?

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