Question 191
Find all positive integers $k$ such that there are finitely many triangles on the Descartes coordinate plane such that
(1) the center of mass of each triangle is an integral point;
(2) the intersection of any two triangles is either the empty set, a common vertex, or an edge joining two common vertices; and
(3) the union of these triangles is a square with side length $k$. (The vertices of the squares are not required to be integral points, and the edges are not required to be parallel to the coordinate axes.)
(1) the center of mass of each triangle is an integral point;
(2) the intersection of any two triangles is either the empty set, a common vertex, or an edge joining two common vertices; and
(3) the union of these triangles is a square with side length $k$. (The vertices of the squares are not required to be integral points, and the edges are not required to be parallel to the coordinate axes.)