CH. Ill] GEOMETRICAL RECREATIONS 55 



in such a way that no two contiguous districts shall be of the 

 same colour. By contiguous districts axe meant districts having 

 a common line as part of their boundaries : districts which touch 

 only at points are not contiguous in this sense. 



The problem was mentioned by A. F. Mobius* in his 

 Lectures in 1840, but it was not until Francis Guthrief com- 

 municated it to De Morgan about 1850 that attention was 

 generally called to it : it is said that the fact had been familiar 

 to practical map-makers for a long time previously. Through 

 De Morgan the proposition then became generally known ; and 

 in 1878 CayleyJ recalled attention to it by stating that he could 

 not obtain a rigorous proof of it. 



Probably the following argument, though not a formal 

 demonstration, will satisfy the reader that the result is true. 



Let A, B, be three contiguous 

 districts, and let X be any other 

 district contiguous with all of them. 

 Then X must lie either wholly out- 

 side the external boundary of the 

 area ABG or wholly inside the in- 

 ternal boundary, that is, it must 

 occupy a position either like X or 

 like X'. In either case there is no 

 possible way of drawing another 

 area Y which shall be contiguous 

 with A, B, G, and X. In other 

 words, it is possible to draw on a 

 plane four areas which are contiguous, but it is not possible 

 to draw five such areas. If A, B, G are not contiguous, each 

 with the other, or if X is not contiguous with A, B, and G, it 

 is not necessary to colour them all differently, and thus the 

 most unfavourable case is that already treated. Moreover any 



* Leipzig Transactions (Math.-phys. Classe), 1885, vol. xxxvh, pp. 1 — 6. 



+ See Proceedings of the Royal Society of Edinburgh, July 19, 1880, vol. x, 

 p. 728. 



t Proceedings of the London Mathematical Society, 1878, vol. ix, p. 148, and 

 Proceedings of the Royal Geographical Society, London, 1879, N.S., vol. i, 

 pp. 259—261, where some of the difficulties are indicated. 



