Avoiding Colinear Points

[Jud McCranie]

At 04:44 PM 10/21/2004, Leroy Quet wrote:

>I describe a sequence and game based on the following idea.
>
>If we have a rectangular grid of n lattice-points by n lattice-points,
>how many dots can we place at the lattice-points, at most one dot per
>lattice-point, so that no 3 or more dots are colinear (in any direction)?
>
>I get the sequence:
>1, 4, 6, 8, 10,..
>And Richard Mathar has extended it:
>12, 14, (15 or 16),...
>
>The upper bound on each term is obviously 2n, since we can have at most 2
>dots in each row  and column.

There is an easy approach to this for nxn.  If you superimpose any two 
different n-queens solutions, you get a solution with no three colinear 
points.  Therefore 2n is always possible (for n > 1).