Instead, interpret the problem as a combinatorial lattice: suppose the triangle is discretized into a triangular grid with $6$ units per side, and points are placed only when coordinates satisfy $x + y + z = 6$, $x,y,z \geq 0$ integers—this is a standard triangular lattice. But the triangle is equilateral in Euclidean sense, embedded in a grid. - Nelissen Grade advocaten