3 lines in a plane can be easily be drawn such that there are 0, 1, 2 or 3 points where at least 2 of them cross.

What are the possible numbers of crossing points for 4, 5, or 6 lines?

Can any of these results be generalized?

(In reply to Other possibilities for 6 lines, and a theorem (unproven) by Steve Herman)

Steve, It is fairly easy to find the number of points for 6 lines. Draw a star (pentagram) with the lines continuing, this gives 10 points for the 5 lines. You can then find where it is possible for 11 points, 12 points, and so forth with the placement of the 6th line.
I agree with your prediction that for n lines in a plane, the number of crossing points can be 0, 1, or any number between n-1 to n(n+1)/2.  Also, as you have predicted, for n lines greater than 3 in the plane, the number of crossing points from 2 to n-2 is not possible. 

Edited on October 11, 2014, 1:21 am

Posted by Dej Mar on 2014-10-10 04:31:12