For 2<=n<=32, it is possible to select 2n lattice points with x, y element [1, n] such that no three are in a straight line (where "straight line" means any line in the plane--not just a horizontal or vertical line). The number of distinct solutions (not counting reflections and rotations) for n = 1, 2, ..., are 1, 1, 4, 5, 11, 22, 57, 51, 156 ... (OEIS A000769). For large n, it is conjectured that it is only possible to select at most (c + ϵ) n lattice points with no three collinear, where c | = | 1/3 πsqrt(3) | ≈ | 1.8138... (OEIS A093602; Guy, pers. comm., Oct. 22, 2004), correcting Guy and Kelly and Guy who found c = (2π^2/3)^(1/3)≈1.87.

integer lattice | N-cluster | tic-tac-toe