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Why is it indeed possible to use just two colors?

Start with the fewest meaningful number of lines. A single line divides the plane into two half-planes; clearly two colors suffice to color the plane: each of the half-planes gets its own color. Note in passing that there are just two possible colorings. One of the half-planes is either white or black whilst the other half is painted with the opposite color.

Now see what happens when you add one line at a time. With each new line we associate an arbitrary manner one of the half-planes it divides the plane into. We'll call it provisionally the line's half-plane. So, when adding a line, leave the colors in its half-plane unchanged but swap all colors in the other half-plane with their opposite colors. Observe that this procedure leads to a two-color coloring of the plane. Indeed, In the line's half-plane where colors have not been changed any two adjacent regions are colored with different colors (for this how they were painted before the last line was drawn.) In the other half-plane this is also true, for changing the colors to their opposites still leaves two adjacent regions painted with different colors (again because this was so before the last line was drawn.) Now, the newly created regions, i.e., the parts of the regions crossed by the last line share a side which is a part of the line; originally, being parts of the same region they were painted with the same color but then the color of one of them has been changed, making them painted in different colors.