How many regions do the sides of a regular $n$-gon divide the plane into?
Solution. The half-planes defined by the sides of a regular polygon always contain the polygon in exactly one of them. Therefore, no three of the side lines can pass through the same point. If any three side lines did pass through the same point, then the polygon would have points in both half-planes defined by the middle line, which is a contradiction. If $n$ is odd, then no two side lines are parallel, but if $n$ is even, then the opposite side lines are parallel.

To determine the number of regions, place the side lines one by one on the plane. Initially, there is 1 region, which increases by 1 when the first line is placed. When placing each new line, the number of regions increases by $(i+1)$, where $i$ is the number of previously placed lines that the new line intersects, because the $i$ intersection points divide the new line into $i+1$ segments, and each segment splits one of the existing regions into two.

![](https://cdn.mathpix.com/cropped/2024_05_02_ce1d44ab4c3cac21e254g-1.jpg?height=309&width=545&top_left_y=602&top_left_x=779)

If $n$ is odd, then the number of intersection points when placing the $i$-th line is $i-1$, so the number of regions increases by exactly $i$, and after placing all the lines, the number of regions is

$$
1+(1+2+3+\ldots+n)=1+\frac{n(n+1)}{2}=\frac{n^{2}+n+2}{2}
$$

If $n$ is even, the situation is different. Let $n=2m$, and first place the first $m$ side lines in some order. None of these lines are parallel to each other, so the $i$-th line has $i-1$ intersection points, and thus the number of regions after placing them is $1+(1+2+\ldots+m)$. When placing the next $m$ lines, each new line will be parallel to exactly one of the previously placed lines. Therefore, the $(m+i)$-th line will have only $m+i-2$ intersection points. In this case, the total number of regions created by all the lines is

$$
1+(1+2+\ldots+m)+(m+\ldots+2m-1)=1+m \cdot 2m=\frac{n^{2}+2}{2}
$$

Thus, the lines divide the plane into the specified number of regions.