15.27. What is the maximum number of parts that $n$ circles can divide a sphere into?
15.27. Let the required number be denoted by $S_{n}$. It is clear that $S_{1} = 2$. Now, let's express $S_{n+1}$ in terms of $S_{n}$. Consider a set of $n+1$ circles on a sphere and single out one of these circles. Let the remaining circles divide the sphere into $s_{n}$ parts ($s_{n} \leqslant S_{n}$), and let the number of parts into which they divide the singled-out circle be $k$. Since $k$ is equal to the number of intersection points of the singled-out circle with the other $n$ circles, and any two circles have no more than two intersection points, then $k \leqslant 2 n$. Each of the parts into which the singled-out circle is divided splits no more than one of the previously obtained parts of the sphere into two parts. Therefore, the $n+1$ circles divide the sphere into no more than $s_{n} + k \leqslant S_{n} + 2 n$ parts, and equality is achieved when any two circles have two common points and no three circles pass through the same point. Thus, $S_{n+1} = S_{n} + 2 n$, and therefore, $S_{n} = S_{n-1} + 2(n-1) = S_{n-2} + 2(n-2) + 2(n-1) = \ldots$ $\ldots = S_{1} + 2 + 4 + \ldots + 2(n-1) = 2 + n(n-1) = n^2 - n + 2$.