Answer: 28.

Solution. Let there be $x$ sides on one side of the diagonal, and $-n-x$ on the other. Then there are $-x-1$ vertices on one side, and $-n-x-1$ on the other. They can be the endpoints of the required diagonals. We get $14=(x-1)(n-x-1)$.

The following cases are possible:

$x-1=14, n-x-1=1$, then $n=17$

$x-1=1, n-x-1=14$, then $n=17$

$x-1=2, n-x-1=7$, then $n=11$;

$x-1=7, n-x-1=2$, then $n=11$.
7. Variant 1.

In a convex $n$-gon, a diagonal is highlighted. The highlighted diagonal is intersected by exactly 14 other diagonals of this $n$-gon. Find the sum of all possible values of $n$. A vertex of the $n$-gon is not considered an intersection.