On the plane, there is a set of points $M$ and seven different circles $C_{1}, C_{2}, \cdots, C_{7}$, where circle $C_{7}$ passes through exactly 7 points in $M$, circle $C_{0}$ passes through exactly 6 points in $M$, ... circle $C_{1}$ passes through exactly 1 point in $M$, then the minimum number of points in $M$ is
(A) 11 ; (B) 12 ; (C) 21 ; (D) 28.
This problem mainly tests students' ability to reason recursively and by induction. Initially, seven different straight lines were used, but later, changing the lines to circles was considered to reduce the problem's intuitiveness, making it more conducive to testing students' recursive and reduction skills. If the points through which the seven circles pass are added without careful analysis, the incorrect answer (D) might be chosen; a slight lack of attention in reasoning could also lead to the wrong choice of (A) or (C). The key lies in carefully and correctly reasoning recursively. To minimize the number of points in set $M$, the points in $M$ on each circle should overlap as much as possible. If circle $C_{7}$ passes through 7 points in $M$, then circle $C_{6}$ can pass through at most 2 points in $M$ on $C_{7}$, meaning $C_{6}$ must pass through at least 4 points in $M$ not on $C_{7}$; $C_{5}$ can pass through at most 2 points in $M$ on $C_{7}$ and 2 points on $C_{6}$, so $C_{5}$ must pass through 1 point in $M$ not on $C_{7}$ or $C_{6}$; $C_{6}, C_{3}, C_{2}, C_{1}$ can pass through the points among the 7 + 4 + 1 = 12 points, without needing to pass through any additional points in $M$, thus the minimum number of points in $M$ is 12, so the correct choice is (B). (Note: The reasoning can also start from circle $C_{1}$ and proceed recursively.)

Furthermore, if the circles in the problem are changed to $n$ circles, with other conditions remaining unchanged, it is not difficult to derive the following recursive formula for the minimum number of points in $M$:
$$
\begin{array}{l}
n+(n-3)+\cdots+\left(n-3\left[\frac{n}{3}\right]\right), \\
n \geqslant 3 .
\end{array}
$$

where $[x]$ is the Gauss (i.e., floor) function.