Putting Dots in Triangles

Given a right-angled triangle of squares in a grid whose horizontal and vertical sides are \( n \) squares long, let \( N(n) \) denote the maximum number of dots that can be placed into the cells of the triangle such that each row, each column, and each diagonal parallel to the long side of the triangle contains at most one dot. It has been proven that \( N_f(n) = \lfloor \frac{2n+1}{3} \rfloor \). In this note, we give a new proof of the upper bound \( N_f(n) \leq \lfloor \frac{2n+1}{3} \rfloor \) using linear programming techniques.