What is the maximum number of regions into which seven circles can divide a plane?

Question

Also, what is the maximum number of regions into which x number of circles can divide a plane?
Explanations please

Answer 1

http://mathworld.wolfram.com/PlaneDivisionbyCircles.html



it is intriguing to notice that this is twice the number by which you can divide a plane by 6 lines.



The formula for lines is straighforward.



http://www.proofwiki.org/wiki/Number_of_Regions_in_Plane_Defined_by_Given_Number_of_Lines



so it would be nice to see how one relate to the other.



edit Take circles of radius 1 all containing some given point inside. That way they all intersect pairwise. Suppose you have n_k regions with the first k circles; Your (k+1)-st circle intersects the previous ones in 2k points which you can assume all distinct by moving your circle as little as you want.

Thus your new circle is divided into 2k arcs, each of them being in one of the regions defined by your first k circles. Each such region is divided in 2 by the new one. That creates 2k new ones.



So x_(k+1) = x_k + 2k. Then you are all set.

Answer 2

Circles , 1 , 2 , 3 , 4 , 5,

regions , 2 , 4 , 8, 14 , 22,

use first and second derivative to find the function

f´(x) .........2...4...6...8...

f´´(x) ..........2...2....2



f´´(x) =2

f´(x) = 2x+k

k=-1

f´(x) = 2x-1

f(x) = x² -x +m

m=2



f(x) = x² -x +2

for 7 circles u have 44 regions

f(7) = 49 -7 +2 = 44

Answer 3

Gianlino's observation is interesting - in fact if Ln is the number of segments created by n lines and Cn the number created by n circles then Cn = 2L(n-1).

Answer 4

Infinity.