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Counting Crossings (Posted on 2014-10-08) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Jer    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: Other possibilities for 6 lines, and a theorem (unproven) Comment 7 of 7 |
(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

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