12. A problem inspired by A.D. Sakharov's problem (7-11). Vasya, in a fit of rage, is cutting a rectangular sheet of paper with scissors. Every second, he cuts the first piece he comes across with a random straight cut into two parts.

a) (1 pt.) Find the expected number of sides of the polygon that will randomly fall into Vasya's hands after an hour of such work.

b) (2 pts.) Solve the same problem if the sheet of paper initially had the shape of an arbitrary polygon.

Note. This problem, in a different formulation and in a more general form, was conceived and solved by Academician Andrei Dmitrievich Sakharov. Sakharov wrote that the problem arose when he was chopping cabbage.
Solution. Let's solve problem b). Suppose that as a result of Vasya's activity, there are $p_{3}$ triangles, $p_{4}$ quadrilaterals, and so on, with the number of $m$-gons being $p_{m}$. Here, $m$ is the largest possible number of sides of the resulting polygon, and it doesn't even matter what $m$ is. Then the expected number of sides of the chosen polygon is

$$
\mu=3 \cdot \frac{p_{3}}{k+1}+4 \frac{p_{4}}{k+1}+\ldots+m \frac{p_{m}}{k+1}=\frac{3 p_{3}+4 p_{4}+\ldots+m p_{m}}{k+1}
$$

Notice that the numerator contains the total number of sides of all the resulting polygons. We need to find exactly this value.

Suppose Vasya had an $n$-gon. The probability of making a random cut exactly through some vertex is zero. Therefore, each cut divides some two sides of the cut polygon into two parts each, and two new sides are formed along the cut. Thus, each cut adds 4 new sides. After the first cut, the total number of sides will be $n+4$, after the second $n+8$, and so on. After $k$ cuts, the total number of sides will be $n+4k$.

Thus, $\mu=\frac{n+4k}{k+1}$.

Notice that if Vasya initially had a quadrilateral, then $\mu=\frac{4+4k}{k+1}=4$, which gives the solution to problem a).

If the initial polygon had a different number of sides $n$, then $\lim _{k \rightarrow+\infty} \mu=4$ and therefore for sufficiently large $k$, the average number of sides can be considered practically equal to 4.