0 lines cuts the plane into at most 1 region.
1 line cuts the plane into at most 2 regions.
2 lines cut the plane into at most 4 regions.
What is the most number of regions that 9 lines can cut the plane into?
Lines Regions
1 2
2 4
3 7
4 11
Using just the regions we have and the sum of differences we have
2 4 7 11
2 3 4
1 1
We have 2 non-zero rows ....so we will have a second power polynomial in the form
an^2 + bn + c and we have these system of equations
a + b + c = 2
4a + 2b + c = 4
9a + 3b + c = 7
Subtract the first equation from each of the other two and we have the system
3a + b = 2 ⇒ -6a - 2b = -4 (a)
8a + 2b = 5 (b)
Add (a) and (b) and we have
2a = 1
a = 1/2
And
8(1/2) + 2b =5
4 + 2b =5
2b =1
b =1/2
And (1/2) + (1/2) + c = 2
1 + c = 2
c =1
So....the resulting polynomial for n lines is
(1/2)n^2 + (1/2)n + 1 =
[ n^2 + n + 2 ]
___________
2
So....for 9 lines we have
9^2 + 9 + 2 92
_________ = ____ = 46 regions
2 2