# Whats the maximum number of parts $n$ circles can break up $\mathbb{R}^2$ into?

Is it $$2^n$$? Think about it, $$1$$ circle breaks up $$\mathbb{R}^2$$ into $$2$$ parts (subsets), such that

1) Any $$2$$ points in each one of the subsets can be connected to one another by a polygonal chain.

2) You can draw a circle with some radius around any point that's an element of such a subset.

If you have $$2$$ circles that intersect, the plane will be broken up into $$4$$ parts.

If you add the third circle, you can have this kind of a picture: So, now, suppose I want to say for $$k$$ circles I have shown that we have $$2^k$$ plane parts. Can I say in the induction step that we can draw another $$k+1$$st circle that will go through all the $$2^k$$ plane parts and thus conclude that it's true that $$n$$ circles can at most break up $$\mathbb{R}^2$$ into $$2^n$$ parts?

I don't understand whether it's always possible to draw a circle that goes through all the plane parts, if you already have $$k$$ circles.

If you have $$k$$ circles, the maximum number of regions into which the plane can be divided is not actually $$2^k$$, rather it is $$k^2 -k + 2$$