The line connecting two points A(n,0) and B(0,n) has a simple equation, x + y = n. Therefore, it contains all the points in the form (i, n-i), i an integer. There are n-1 such points between A and B. Connect each one of them with the origin O. The lines divide OAB into n small triangles.
Superimpose an integer grid on the diagram. The question is about the number of grid points that lie inside small triangles. Obviously, the two triangles next to the axes contain no grid points in their interiors. Prove that, for n prime, each of the remaining triangles contains exactly the same number of grid points.
Now note that small triangles have the same base (1/n-th of AB) and the same height. Thus all of them share the same area. Since each of the triangles in question has exactly 3 grid points on its boundary (at its vertices), the conclusion follows immediate from Pick's theorem.
From Pick's theorem, n2 = I - B/2 + 1, where I is the number of grid points in the interior of the square and B is the number of points on its boundary. The total number of points covered by the square is given by I + B:
I + B = I + B/2 - 1 + B/2 + 1 = n2 + B/2 + 1 n2 + 4n/2 + 1 = (n + 1)2