A point in the plane is said to be a lattice point if it has integral coordinates. Given a set C of lattice points and two non-negative integers m and n, we are concerned with determining a set of m vertical and a set of n horizontal straight lines so that the number of intersections of these sets is maximal. In general, this problem is rather difficult, since we have to make simultaneous decisions about the positions of the vertical and horizontal straight lines. Our purpose here is to study a set of lattice points of the form (i, j) such that i, j, i + j and i − j are bounded by fixed integers. We shall show that, in this case, sequential decisions can be made about the two families of straight lines.