## Task 19/70

A prism with an $n$-sided base has 10n diagonals (body and face diagonals). What is $\mathrm{n}$?
A $n$-sided prism has $2n$ vertices and $3n$ edges. Each vertex can be connected to any other vertex by a line segment that is either an edge or a diagonal. The number of connecting segments is

$$
\frac{2n(2n-1)}{2}=n(2n-1)=2n^2-n
$$

Therefore, the number of diagonals is $10n=2n^2-n-3n$. This results in the quadratic equation without a constant term: $n^2-7n=0$ with the solutions $n_1=0, n_2=7$. Since there is no zero-sided prism (the domain for $n$ is $n \geq 3$, whole!), $n=7$ is the only solution. A heptagonal prism thus has a total of 70 diagonals.