How many colors?
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HOW MANY COLORS?
On the cover of the magazine you see four circles, each of which contains a chord. They divide the plane into several parts. These parts are painted with three colors so that the neighboring ones have different colors. (Parts whose boundaries have a common segment or arc are called neighboring. Parts whose boundaries contain one common point are not considered neighboring.) We call such a coloring correctly, and a point that belongs to the boundary of more than two parts is a vertex.
Prove that if the plane is divided into parts:
a) several washed, but for correct coloring there are two colors available;
b) circles, in each of which a chord is drawn, but two colors are available for correct coloring. (In the last problem, the plane is divided by arbitrary lines.)
In these problems, for correct coloring it is enough to correctly color an arbitrary plane into parts, when it is no longer possible to bypass it with three colors. Make sure that in each of your examples of four colors to achieve enough colors, proven.
You can read more about this in the books by G. Rademacher and O. Tellitz “Numbers and Figures”, M., “Science”, 1966 and L. I. Golovina and I. M. Yaglom “Induction and Geometry”, M., Gostekhidat, 1956.