For nearly 80 years, mathematicians have studied a deceptively simple question: if you place nn points in the plane, how many pairs of points can be exactly distance 11 apart?This is the planar unit distance problem, first posed by Paul Erdős in 1946. It is one of the best-known questions in combinatorial geometry, easy to state and remarkably difficult to resolve. The 2005 book Research Problems in Discrete Geometry, by Brass, Moser, and Pach, calls it “possibly the best known (and simplest to ...