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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?

 Nov 18, 2019
 #1
avatar+105370 
+1

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

 

 

 

cool cool cool

 Nov 18, 2019

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