button to main menu  Four Colours?


Guthrie 1850-52

At a lecture given by Agustus de Morgan, professor of mathematics, University College, London, 23 October 1852, one of the students, Frederick Guthrie, asked a simple question. De Morgan mentioned the suggested conjecture in a letter to Sir William Rowan Hamilton, mathematician and physicist; was it a fact that:-
... if a figure be divided and the compartments differently coloured so that figures with any portion of common boundary line are differently coloured - four colours may be wanted but not more. ...
The conjecture was first proposed by Frederick's elder brother Francis Guthrie, an earlier student of de Morgan's at University College, graduated 1850, later a professor of mathematics in South Africa.
This then is the:-

Four Colour Theorem

Every map drawn on a plane can be coloured with at most four colours in such a way that neighbouring countries are coloured differently.
Some maps can be coloured with fewer colours.
Maps having countries with detached parts are not included in the scope of the theorem. Countries must be in one piece.
Where more than three countries meet at a point the point is not regarded as a boundary; boundaries are boundary lines.


The four colour theorem does not arise out of practical cartography. According to Kenneth May, mathematical historian who studied a sample of atlases in the Library of Congress, there is:-
no tendency to minimise the number of colors used. Maps utilizing only four colors are rare, and those that do usually require only three.
Textbooks on cartography and the history of cartography don't mention the four colour theorem, even though map colouring is a subject of discussion. The four colour theorem has no origin in cartography.
I talked about the issue to two well known practising cartographers at the British Cartographic Society meeting 2003, asking how they set about colouring a map. Both said they just got on a did it, trial and error; using computer mapping software it was very quick to experiment. Both were much more concerned that colours were balanced, that no colour dominated, and that there were not too many colours, which would not be clearly recognised.

May, Kenneth O: 1965: Origin of the Four Color Conjecture: Isis: vol.56: pp.346-348

Coxeter, H S M: 1959: Four-Color Map Problem 1840-1940: Mathematiclal Teacher: vol.52: pp.283-289

Four Colours Suffice

After many faulted attempts the theorem was proved by Kenneth Appel and Wolfgang Haken, publicly announced 22 July 1976, and published 1977:-

Appel, Kenneth & Haken, Wolfgang & Koch, John: 1977 (December): Every Planar map is Four Colorable: Illinois Journal of Mathematics: vol.21: pp.439-567
There is a more approachable article, not a proof:-

Appel, Kenneth & Haken, Wolfgang: 1977 (October): Solution of the Four Color Map Problem: Scientific American: vol.237 no.4: pp.108-121
The proof of the four colour theorem is not simple; it involves more than a 1000 hours of computing time (1977 computer) checking more than 100000 particular cases. To some mathematicians this is unacceptabe; the proof cannot be reviewed: but would a 1000 page hand written proof be any more checkable?
What the proof does not provide is any insight as to why the conjecture is true. The theorem is true, but unexplained. The proof is terrible, inelegant mathematics.

button to main menu Lakes Guides menu.