It is easily visible for $n=1,2,3$ that $n$ circles can divide the plane into at most $2^{n}$ parts. What is the maximum number of parts the plane can be divided into by drawing four circles, by appropriately choosing the size and relative positions of the circles?
Suppose we have already drawn three circles, and these divide the plane into $k$ parts. Consider an arbitrary fourth circle and assume that it has $l$ points of intersection with the previous ones.

If $l=0$, then four circles divide the plane into $k+1$ parts, and if $l>0$, then the plane is divided into at most $k+l$ parts. The first statement is obvious, and the second follows from the fact that if we sequentially remove the arcs of the fourth circle that lie between two consecutive intersection points, the number of plane regions decreases by at most one at each step (specifically, the regions on either side of the arc merge, but these regions are not necessarily distinct, as shown in Figure 1, where the removal of $\widehat{A B}$ followed by the removal of $\widehat{C D}$ does not increase the number of regions). The number of arcs on the fourth circle is exactly the number of intersection points, which is $l$.

![](https://cdn.mathpix.com/cropped/2024_05_02_bcbe9ac6fa1e4d5f452eg-1.jpg?height=323&width=415&top_left_y=519&top_left_x=839)

Figure 1

The fourth circle can intersect the previous three at most in 6 points, and three circles can divide the plane into at most 8 parts, so four circles can create at most 14 parts. This is achievable, as shown in Figure 2.

![](https://cdn.mathpix.com/cropped/2024_05_02_bcbe9ac6fa1e4d5f452eg-1.jpg?height=257&width=355&top_left_y=1075&top_left_x=869)

Figure 2

Remark. From our reasoning, it also follows that the $(n+1)$-th circle can create at most $2n$ new plane regions after $n$ circles. Thus, if $f(n)$ denotes the maximum number of plane regions created by $n$ circles, then (since one circle divides the plane into 2 parts)

$$
f(n) \leqq 2+2+4+6+\cdots+2(n-1)=n(n-1)+2
$$

It can also be shown that $f(n)=n(n-1)+2$.