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2007 intersections (Posted on 2007-02-01) Difficulty: 3 of 5
Each of N lines on a plane intersects exactly 2007 other lines. Find all possible N.

No Solution Yet Submitted by Art M    
Rating: 3.6667 (3 votes)

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I think | Comment 1 of 4
2008 ≤ N ≤ 4014

If none of the lines are parallel, 2008 is enough. It's just geometry, really. The more parallel the lines are, the more you need, leading up to 4014.

But here's a more intuitive explanation:
The upper limit is obvious - if you place 2007 parallel lines and another 2007 at a right angle to the first ones, you'll get a nice grid. Any more added will be too much.
The lower limit is slightly less obvious. If you draw a triangle (3 lines, each one intersecting 2 others), then you can add another line that will pass through all of the first three, making it 4 lines, with each intersecting 3. This process can be repeated, and allows for N+1 lines to each intersect N lines.

As for all of the options in the middle, I'm just guessing. I don't know if they're all possible. Someone help me? ^_^

EDIT: Actually, now I'm pretty certain that not all of the middle options are possible.

Edited on February 1, 2007, 8:55 am
  Posted by TamTam on 2007-02-01 08:35:20

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