Crazy lines

Geometry Level 3

Six straight lines are drawn in a plane with no two parallel and no three concurrent. Find the number of regions into which they divide the plane.


The answer is 22.

Naitik Sanghavi by naitik sanghavi , 20, India

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1 solution

Partho Kunda
Jul 4, 2015

For this problem we have to use recursion method.
Suppose, for n lines the number of regions are L n L_{n} .
If there is no line in the page. Then L 0 L_{0} =1.
And L 1 L_{1} =2,
L 2 L_{2} =4=2+2= L 1 L_{1} +2,
L 3 L_{3} =7=4+3= L 2 L_{2} +3,
L 4 L_{4} =11=7+4= L 3 L_{3} +4.
So, L n L_{n} = L n − 1 L_{n-1} +n = L n − 2 L_{n-2} +n-1+n = L 0 L_{0} +1+2+.......+n-1+n
=1+ n ( n + 1 ) 2 \frac{n(n+1)}{2}
Thus, L 6 L_{6} =1+21=22


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