## Spring 2016, problem 15

A lattice point is defined as a point in the $2$-dimensional plane with integral coordinates. We define the centroid of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, as the point $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Show that $n = 12$.