Four. (This question is worth 50 points)
On a plane, there are $4 n\left(n \in N^{*}\right)$ circles of the same radius, where no two circles are tangent, and any one circle intersects with at least three other circles. Let the number of intersection points of these circles be $f(n)$, find the minimum value of $f(n)$.
Four, Solution: The minimum value $f_{\min }(n)=4 n$.
Let $M$ be the set of these $4 n$ circles $C_{1}, C_{2}, \mathrm{~L}, C_{4 n}$, and $N$ be the set of the intersection points $P_{1}, P_{2}, \mathrm{~L}, P_{f(n)}$ of these $4 n$ circles. For a circle $C \in M$ and an intersection point $P \in N$, if point $P$ is not on circle $C$, define $F(C, P)=0$; if point $P$ is on circle $C$, define $F(C, P)=\frac{1}{m}$, where $m$ is the number of circles passing through point $P$. Then for any point $P$ in $N$, we have $\sum_{i=1}^{4 n} F\left(C_{i}, P\right)=1$. For any circle $C$ in $M$, let $\frac{1}{m}=F(C, P)=\min _{1 \leq j \leq f(n) \Perp P, \text { on } C }\left\{F\left(C, P_{j}\right)\right\}$, and denote $C_{1}^{p}, C_{2}^{p}, \mathrm{~L}, C_{m-1}^{p}$ as the other $m-1$ circles passing through point $P$, where the other intersection point of circle $C_{i}^{p}(i=1,2, \mathrm{~L}, m-1)$ and circle $C$ is $Q_{i}$, and these circles have the same radius, then the $m-1$ points $Q_{i}$ are all distinct.
$$
\begin{array}{l}
\therefore \sum_{j=1}^{f(n)} F\left(C, P_{j}\right) \geqslant F(C, P)+\sum_{i=1}^{m-1} F\left(C, Q_{i}\right) \geqslant \frac{1}{m}+(m-1) \frac{1}{m}=1 \\
\therefore f(n)=\sum_{j=1}^{f(n)} \sum_{i=1}^{4 n} F\left(C_{i}, P_{j}\right)=\sum_{i=1}^{4 n} \sum_{j=1}^{4 f(n)} F\left(C_{i}, P_{j}\right) \geqslant=\sum_{i=1}^{4 n} 1=4 n
\end{array}
$$

Next, we show that $f(n)=4 n$ is achievable,
As shown in the figure: every four circles have 4 intersection points, (since all circles have radius $r$, the side length of the equilateral $\triangle P Q R$ is $\sqrt{3} r$, these four circles are the circumcircles of $\triangle P Q R, \triangle S P Q, \triangle S Q R, \triangle S R P$) $\therefore f_{\min }(n)=4 n$.
$$
\begin{array}{lll}
\hline Q & P S & R
\end{array}
$$