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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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Solution re: I think - therefore I am - I think | Comment 2 of 4 |
(In reply to I think by TamTam)

2008 and 4014 look good.

I would only add 2010, 2016, 2230, and 2676.

I expanded on your idea of groups of parallel lines, but instead of 2 groups, I used 4, 10 and 224. Imagine 4 groups of parallel lines intersecting. Since they intersect all of the lines of the other groups, those groups would have to be made up of 2007/3 lines each = 669. Since there are 4 groups the total is 669 * 4 = 2676

The factors of 2007 are {1, 3, 9, 223, 669, and 2007}.   If each of the sets of parallel lines is made of that number of lines (m), then there can be 2007/m + 1 sets of parellel lines, and (2007/m + 1)*m lines. The corresponding total number of lines would be {2008, 2010, 2016, 2230, 2676, and 4014}

Basic structure would be n = 2007 + 2007/m where m (and n) are integers.

Edited on February 1, 2007, 9:25 am
  Posted by Leming on 2007-02-01 09:16:41

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