Question 164
Consider a rectangular board of $m \times n$ cells with $m, n \ge 1$. The vertices of the cells form a $(m+1) \times (n+1)$-grid. We say a triangle whose vertices are points on the grid is *low* if there is at least one side of the triangle that is parallel to a side of the board and for which the height of the triangle with this side as its base equals 1. We say a *low triangle* is *special* if there are two sides that are parallel to a side of the board. We partition the board in low triangles.
Determine the minimum number of special triangles over all possible partitions of the $m \times n$-board.
Determine the minimum number of special triangles over all possible partitions of the $m \times n$-board.