Given a natural number $n$, consider the following 2-player game: at each turn, a player places an (infinite) line on the plane (that wasn't already placed). The first player wins if at any point in the game there is an $n$-gon on the plane. Otherwise, the second player wins. By "there is an $n$-gon on the plane", I mean that there is an $n$-gon whose side are lines placed by the players, and that has no lines inside of it.
For what $n$ does the first player have a winning strategy?
Checking "by hand", it's easy to see that for small $n$ (up to about $n=5$) the first player can force a win, but I can't come up with a general strategy. It also feels counter intuitive for the first player to have a winning strategy for very large $n$, since for example the second player can split a $100$-gon into $2$ $51$-gons in a single move, so to get to $101$ there would have to be multiple $100$-gons made at the same time.
The strategy for $n=3$: the first player places a line. If the second player places a line that is not parallel to it, the first player can immediately create a triangle. If it is parallel, the first player places a line perpendicular to the 2 existing lines, and will be able to create a triangle on their next turn regardless of what the second player does.
For $n=4$: notice a line can never "uncreate" a triangle, only turn it into 2 smaller triangles or create a quadrilateral. So the first player creates a triangle with the first strategy, and then makes it into a quadrilateral on their next turn.
For larger $n$ the strategies are similar to the first one, but with a lot more cases.