# How to divide plane with four circles to get Maximum number of region [duplicate]

I started to divide flat plane with one circle to get maximum number of regions and I got 2. Then I tried to do this with two circles And I got 4 different regions. Then I did this with tree circles and I got 8 different regions. After that I thought that with four circles I will get 16 different regions. Anyway I tried to divide flat plane with four circles to get 16 different regions but maximum number of that regions were 14 not 16! Does anyone have idea what is a relationship between number of circles and maximum number of regions? I thought that the relationship was 2^n where n is number of circles. but if n = 4 this relationship is incorrect!

------------------------------------------------------------------------------------------------------------- it is easy to see for one circle

as we can see if we are trying to divide flat plane with two circles, those circles must have intersection to get maximum regions

for three circles there is many possibility to place those circles on the flat plane.

also there is same situation for three circles!

but as you can see I can not place four circles to get more than 14 regions! why?

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Do you want all circles of same size? –  Aang Mar 5 '13 at 11:30
See Why can a Venn diagram for 4+ sets not be constructed using circles?. The simplest explanation is from Wiki Answers: "Since the fourth circle intersects the first three in at most 6 places, it creates at most 6 new regions; that's 14 total". –  Rahul Mar 5 '13 at 11:32
As for the question in the title, see Counting number of distinct regions with intersecting circles. The formula is $n^2-n+2$. –  Rahul Mar 5 '13 at 11:36
Till number of circles 3, when you add another circle, it can intersect all the regions till created and hence it doubles the number of region every time till you reach fifth circle. When you add fourth circle, it can cut atmost 6 existing regions and hence number of regions increses by 6 only, and if you add fifth circle, it can cut atmost 8 regions and thus giving 22 regions. If you go further, the number of new regions added grows very slowly as compared to $2^n$. It's very less from that.I guess it's polynomial. –  Aang Mar 5 '13 at 11:41