TITLE: Do circles divide the plane into more regions than lines?
QUESTION [54 upvotes]: In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. 
What happens if we use circles instead of lines? That is, what is the maximum number of regions into which n circles can divide the plane?
After some exploration it seems to me that in order to get maximum division the circles must intersect pairwise, with no two of them tangent, none of them being inside another and no three of them concurrent (That is no three intersecting at a point).
The answer seems to me to be affirmative, as the number I obtain is $n^{2}-n+2$ different regions. Is that correct?

REPLY [8 votes]: A qualitative approach:
Consider a group of $n$ lines that divide the plane into maximally many regions. The intersections between the lines all fall within a bounded area of the plane. Now replaces each of the lines with a huge circle, so large that within the bounded area-of-intersections it deviates from a straight line by much less than the distance between two points of intersection.
Then you still have all the regions you had with lines. However, if there are at least three circles, you also get some new regions that are far away from the central bounded area.