## Task $8 / 68$

A plane is divided into 56 parts by $n$ lines. None of the $n$ lines is parallel to another, and no more than two lines intersect at any point.

What is $n$?
Claim: Under the conditions mentioned in the problem, $n$ lines divide the plane into

$$
k_{n}=1+\frac{n(n+1)}{2}
$$

parts.

Proof: The claim is obviously true for $n=0$ (the plane is not divided, so it consists of one part) and for $n=1$ (the plane is divided once, thus into two parts).

Assume the claim is true for $n=i$: $k_{i}=1+\frac{i(i+1)}{2}$.

Then the $(i+1)$-th line creates $i+1$ plane parts. Since this line is not parallel to any other, it intersects each of the others; since it also does not pass through any intersection point of two lines, it intersects $i-1$ plane parts between the $i$ lines and 2 parts outside of them, thus a total of $i+1$ plane parts into 2 parts each. Therefore,

$$
k_{i+1}=1+\frac{i(i+1)}{2}+i+1=1+\frac{(i+1)(i+2)}{2}
$$

Thus, the validity for $n=i$ implies the validity for $n=i+1$. This shows that the claim is true overall.

From the equation $k_{n}=1+\frac{n(n+1)}{2}=56$ it follows that $n=10$.

(Since $n$ is a natural number, the negative solution of the resulting quadratic equation is excluded.) Therefore, there are 10 lines in the plane.