If two lattice points in the plane are chosen and joined by a line segment, the midpoint of this segment may or may not also be a lattice point.
Suppose you try selecting a set of points so that for every pair, the midpoint is not a lattice point.
Is there a limit to how many points you can pick?
Since the line Y=X passes through an infinity of lattice points and the mid-point of any two of these is an integral or half-integral multiple of sqrt(2), the answer is no, there is no limit.
Point A is (a,a), point B is (b,b), midpoint is (1/2)*(b-a)*sqrt(2).
Posted by xdog
on 2017-08-27 18:54:18