Dudeney’s No-Three-In-Line Problem
What is the maximum number of points in an n x n grid such that no 3 points lie on a line? Here we mean not just vertical, horizontal, and diagonal lines, but lines of any slope. Dudeney introduced this simple sounding problem in 1917, but it is still mysterious today.
We will discuss bounds for this problem and a conjectural answer due to Guy and Kelly. Along the way we will see ideas from number theory and algebraic geometry, and many computational and algorithmic directions to explore.