58 GEOMETRICAL RECREATIONS [CH. Ill 



First, suppose that the boundary lines of contiguous dis- 

 tricts form a closed network of lines joining an even number 

 of points such that three and only three lines meet at each 

 point. Then if the number of districts is n + 1, the number of 

 boundaries will be 3n, and there will be 2n points of junction; 

 also by Tait's theorem, the boundaries can be marked with three 

 colours /S, 7, 8 so that no two like colours meet at a point of 

 junction. Suppose this done. Now take four colours, A,B,G, D, 

 wherewith to colour the map. Paint one district with the colour 

 A; paint the district adjoining A and divided from it by the 

 line /8 with the colour B ; the district adjoining A and divided 

 from it by the line 7 with the colour G; the district adjoining A 

 and divided from it by the line 8 with the colour D. Proceed 

 in this way so that a line /3 always separates the colours A 

 and B, or the colours G and D; a line 7 always separates 

 A and G, or D and B; and a line 8 always separates A and D, 

 or B and G. It is easy to see that, if we come to a district 

 bounded by districts already coloured, the rule for crossing each 

 of its boundaries will give the same colour : this also follows 

 from the fact that, if we regard /3, 7, 8 as indicating certain 

 operations, then an operation like 8 may be represented as 

 equivalent to the effect of the two other operations /3 and 7 

 performed in succession in either order. Thus for such a map 

 the problem is solved. 



In the second case, suppose that at any point four or more 

 boundaries meet, then at any such point introduce a small 

 district as indicated below: this will reduce the problem to 

 the first case. The small district thus introduced may be 



* * 



coloured by the previous rule; but after the rest of the map is 

 coloured this district will have served its purpose, it may be 

 then made to contract without limit to a mere point and will 

 disappear leaving the boundaries as they were at first. 



Although a proof of the four-colour theorem is still wanting, 

 no one has succeeded in constructing a plane map which requires 



