37. Find all natural numbers $k$ that have the property that there does not exist a $k$-gon such that the extension of any of its sides contains another side of this $k$-gon. Only polygons with non-parallel adjacent sides are considered.
74.37. Answer. These are all even numbers greater than nine, and all odd numbers greater than 14.

Indeed, consider an arbitrary line on which lie at least two sides $AB$ and $CD$ of the given $k$-gon. Then there are at least four more lines on which lie the sides of the $k$-gon - specifically those sides whose one endpoint is one of the points $A, B, C,$ and $D$ (but not $AB$ and not $CD$). Therefore, there are at least five such lines and at least ten sides.

If $k$ is odd, then on one of these lines there must lie at least three sides of the $k$-gon, which implies that there are at least seven lines, i.e., $k \geqslant 15$.

Finding specific examples of such $k$-gons is left to the reader as an exercise.