n circles and 1 straight line - divide into regions
Original question:
There are n circles and 1 straight line of a plane such that you can divide the plane into at most 44 parts. Find n.
So I have no idea how to do this question, the word ‘plane’ is not defined clearly.
What I mean is that, for example, refer to the picture below. Does the circle in the picture divide the plane into 1 region or 2 regions?
1 Answer
$\begingroup$Consider $n$ circles and a line on a plane. Using a stereographic projection whose projection point is on the line but not any of the circles, we obtain $n+1$ circles on a sphere. Picking another projection point not on any of these $n+1$ circles, we get $n+1$ circles on the plane. Thus your problem is equivalent to calculating the maximal number of regions $n+1$ circles can divide a plane into. For that problem, you can check out plane division by circles.
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