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?
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.
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Posted by Charlie
on 2014-10-08 13:12:45 |
solution
