30 The four-colour problem
Who might have given young Tiny Tim a Christmas present of four coloured wax crayons
and a blank county map of England? It could have been the cartographer neighbour who
occasionally sent in small gifts, or that odd mathematician Augustus De Morgan, who
lived nearby and passed the time of day with Tim’s father. Mr Scrooge it was not.
The Cratchits lived in a drab terrace house in Bayham Street, Camden Town just north of
the newly opened University College, where De Morgan was professor. The source of the
gift would have become clear in the new year when the professor called to see if Tim
had coloured the map.
De Morgan had definite ideas on how this should be done: ‘you are to colour
the map so that two counties with a common border have different colours’.
‘But I haven’t enough colours’, said Tim without a thought. De Morgan would
have smiled and left him to the task. But just recently one of his students,
Frederick Guthrie, had asked him about it, and mentioned a successful colouring
of England with only four colours. The problem stirred De Morgan’s
mathematical imagination.
Is it possible to colour any map with just four colours so that the regions are
distinguished? Cartographers may have believed this for centuries but can it be
proved rigorously? We can think of any map in the world besides the English
county map, such as American states or French départements, and even artificial
maps, made up of arbitrary regions and borders. Three colours, though, are not
enough.