I am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese)
A plane is divided by many lines. Show that it is possible to color the regions formed with only two colors so that no two adjacent regions share the same color.
First of all, I was not able to solve it. Then I did I search on Google, and I've find the following here.
4.Show that if n lines are drawn on the plane so that none of them are parallel, and so that no three lines intersect at a point, then the plane is divided by those lines into $\dfrac{n^{2} + n + 2}{2}$ regions.
The next exercise is:
- Show that if the same lines as in problem $4$ are drawn on a plane that it is possible to color the regions formed with only two colors so that no two adjacent regions share the same color.
Proof: With zero lines, you can obviously do it; in fact, one color would be sufficient. If you can successfully $2$-color the plane with $k$ lines, when you add the $(k + 1)$st line, swap the colors of all the regions on one side of the line. This will provide a $2$-coloring of the configuration with $k + 1$ lines. (In fact, for this problem, there is no real need to have the lines in general position: some can be parallel, and multiple lines can pass through a point, and the proof will continue to work.)
I did a drawn, and I got convinced, but I did not understand it. I am feeling stupid, but I was not able to understand the proof without drawing.
I would appreciate your help.