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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:

enter image description here

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.

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See this link on Plane Division by 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 $$k^2 -k + 2$$

The maximum number of regions, as a function of the number of circles used, is the sequence A014206 on the Online Encyclopedia of Integer Sequences.

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    $\begingroup$ Oh, wow, thanks! $\endgroup$ – Coder-Man Sep 25 '18 at 21:19

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