# 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?

• 2 regions: the region inside the circle and the region outside the circle. May 11, 2020 at 8:17
• I would say 2 regions. In your problem, there will be two unbounded parts, I think. May 11, 2020 at 8:18

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.
• As an aside, we could perform inversion about a point not on the circles or lines, to conclude that it's equivalent to calculating maximal number of regions $n+1$ circles can divide the plane into. Nov 19, 2020 at 3:00