Question 174
Consider a rectangular board of $m \times n$ cells with $m, n \ge 1$. The vertices of the cells form an $(m+1) \times (n+1)$-grid. We call a triangle whose vertices are points of the grid *low* if it has at least one side that is parallel to a side of the board such that the height of the triangle on that side is 1. We call a low triangle *special* if it has two sides that are parallel to sides of the board. We partition the board into 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.