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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)

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Solution solution | Comment 2 of 7 |
Each added line, to make n lines in all, will in general cross all n-1 of the previous lines. That gives a sequence of:

n   max (general) number of crossings
1    0
2    1
3    1+2=3
4    1+2+3=6
5    1+2+3+4=10
6    1+2+3+4+5=15

The maximum number of crossings is just the sequence of triangular numbers: T(n-1).

Any number below that is possible, as 1, 2, ... n-1 can be made parallel to some chosen line.

So in general any number from 0 to T(n-1) of crossing points can exist.

  Posted by Charlie on 2014-10-08 13:12:45
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