Connecting $1997$ points in the plane- what am I missing?

Question

I know I am missing something in this problem, but I don't know what:

Let $P_1, P_2, ..., P_{1997}$ be distinct points in the plane. Connect the points with the line segments $P_1P_2, P_2P_3, P_3P_4, ..., P_{1996}P_{1997}, P_{1997}P_{1}$. Can one draw a line that passes through the interior of every one of these segments?

According to how I understand the problem, the solution is easy. It's clearly not possible to have one line pass through every line segment in this picture.

Answer 1

Consider such a collection of points ${<}P_i{>}$ as described with $N$ points.

Now let us assume that a line can be drawn crossing each of the line segments ${<}L_i{>}$ where $L_N$ connects $P_N$ and $P_{1}$ and all other $L_i$ connect $P_i$ and $P_{i+1}$.

Now we know that if $P_i$ is one one side of the line, $P_{i+1}$ is on the other side of the line, and $P_1$ and $P_N$ are on opposite sides of the line. We can thus infer that $N$ is even, since all odd points are on the same side of the line as $P_1$.

Thus when such a line can be drawn, $N$ must be even, and by contrapositive, if $N$ is odd, such a line cannot be drawn.