Do there exist 100 distinct lines in the plane having just 1985 distinct points of intersection?
Solution
Solution by Demetres Christofides
Answer: yes.
Take 73 lines parallel to the x-axis and 26 lines parallel to the y-axis. These meet in 73.26 = 1898 points. Now take the 73 lines as y = 1, 2, ... , 73 and the 26 lines as x = 1, 2, 3, 4, 5, 6 and 101, 102, ... , 120. Take the 100th line as y = x. It passes through 6 of the existing points of intersection (namely (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) and (6, 6) ). So it only creates (73 + 26 - 12) = 87 new points of intersection, giving 1985 in total.
© John Scholes
jscholes@kalva.demon.co.uk
11 Sep 2002