23. In a plane, what is the maximum number of regions that can be created by $n$ circles? In space, what is the maximum number of regions that can be created by $n$ spheres?
23. (1) Let $k$ circles divide the plane into at most $f_{k}$ regions. When adding the $(k+1)$-th circle, there are $2k$ intersection points, which divide the $(k+1)$-th circle into $2k$ segments. Each segment divides one of the original regions into two. Therefore, $f_{k+1}=f_{k}+2k$. This leads to the formula $f(n)=n^{2}-n+2$.
(2) Let $k$ spheres divide the space into $F_{k}$ regions. When adding the $(k+1)$-th sphere, to maximize the number of regions, it must intersect with all the previous $k$ spheres. This results in $k$ circles on the surface of the $(k+1)$-th sphere, and these $k$ circles intersect pairwise, dividing the sphere into $k^{2}-k+2$ regions (similar to the discussion of $k$ circles dividing the plane).
Thus, $F_{k+1}=F_{k}+k^{2}-k+2$, which leads to the formula $F_{n}=\frac{n\left(n^{2}-3 n+8\right)}{3}$.