Induction in Geometry (Little Mathematics Library) by LI et al Golovina

By LI et al Golovina

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Let us show (a) (c) (b) FfG. I~ that in this case the map has the form illustrated in Fig. 18, b or c. Let us denote the number of map bigons by k 2 , the number of its triangles by k 3 , and the number of quadrilaterals by k 4 (since p = 4~ it is impossible for a map to have countries whose * Here and elsewhere we are not going to distinguish between 'equally constructed' maps (as ilJustrated in Figs. 14, C and 26, a)~ whose countries and boundaries can be numbered so that in both maps equally numbered countries are separated by equally numbered boundaries.

Consider the three possible cases. 56 (a) The vertex A o is joined (by one, two or three boundaries) to only one vertex A 1 of the map 5' (Fig. 38, Q, b, c). In this case the numbering of the boundaries of the map S' is readily extended to a proper numbering of the map S. (b) The vertex A o is joined to two vertices A 1 and A 2 of the map S', with one of which it may be connected with t\VO boundaries (Fig. 39, a and b). It is easy to verify that in all cases the numbering of the boundaries of the map S' can be extended to a proper numbering of the boundaries of the map S.

Frederick did not know, and he asked De Morgan, the well-known professor at Cambridge University, who did not know either. But De Morgan made this problem popular among his colleagues. Thus, both the four-colour problem and the basic method of proof have deep roots in mathematics. In 1878 the leading English mathematician, Arthur Cayley, unable to prove or disprove the four-colour conjecture, presented the problem to the London Mathematical Society. At the end of his report he invited everybody present to take part in solving this problem, thus "letting the genie out of the bottle", The four34 colour conjecture became a really famous problem.

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