Points with integer co-ordinates, like (0, 4) or (-3, 7) are called lattice points.

The centroid of a triangle is the point obtained by finding the mean of the co-ordinates of the vertices of the triangle.

So if I were to choose the three lattice points (2, 1), (4, 7) and (3, -2) as the vertices of a triangle, the centroid would be (3, 2), because the mean of 2, 4, and 3 is 3, and the mean of 1, 7 and – 2 is 2.

Given any set of nine lattice points, can you prove that it will always be possible to pick three of them so that the centroid of the triangle they form is also a lattice point?

Thanks to Katherine in Y13 for providing this challenge.