There arenunit circles andnlines drawn in the plane. Prove that th.pdf

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There arenunit circles andnlines drawn in the plane. Prove that the regions in the plane separated by the lines and circles can be colored with two colors in such a way that no two adjacent regions have the same color. There arenunit circles andnlines drawn in the plane. Prove that the regions in the plane separated by the lines and circles can be colored with two colors in such a way that no two adjacent regions have the same color. Solution We prove it by induction in the number n of lines. For n = 1 we will have two regions, and we can color them with just two colors, say one in red and the other one in blue. Next assume that the regions obtained after dividing the plane with n lines can always be colored with two colors, red and blue, so that no two regions that share a boundary have the same color. We need to prove that such kind of coloring is also possible after dividing the plane with n + 1 lines. So, the plane divided by n lines has been colored in the desired way. After we introduce the (n + 1)th line we need to recolor the plane to make sure that the new coloring still verifies that no Two regions that share a boundary have the same color. We do it in the following way. The (n + 1)th line divides the plane into two half-planes. We keep intact the colors in all the regions that lie in one half-plane, and reverse the colors (change red to blue and blue to red) in all the regions of the other half-plane. So if two regions share a boundary and both lie in the same half-plane, they will still have different colors. Otherwise, if they share a boundary but are in opposite half- planes, then they are separated by the (n + 1)th line; which means they were part of the same region, so had the same color, and must have acquired different colors after recoloring..

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