the eminent Cambridge mathematician Arthur Cayley to write a paper on it in
- Unhappily, Cayley was forced to admit that he had failed to obtain a proof,
but he observed that it was sufficient to consider only cubic maps (where exactly
three countries meet at a point). This contribution spurred on his student Alfred
Bray Kempe to attempt a solution. Just one year later Kempe announced he had
found a proof. Cayley heartily congratulated him, his proof was published, and
he gained election to the Royal Society of London.
What happened next?
Kempe’s proof was long and technically demanding, and though one or two
people were unconvinced by it, the proof was generally accepted. There was a
surprise ten years later when Durhambased Percy Heawood found an example of
a map which exposed a flaw in Kempe’s argument. Though he failed to find his
own proof, Heawood showed that the challenge of the four-colour problem was
still open. It would be back to the drawing boards for mathematicians and a
chance for some new tyro to make their mark. Using some of Kempe’s
techniques Heawood proved a five-colour theorem – that any map could be
coloured with five colours. This would have been a great result if someone could
construct a map that could not be coloured with four colours. As it was,
mathematicians were in a quandary: was it to be four or five?
The simple donut or ‘torus’
The basic four-colour problem was concerned with maps drawn on a flat or
spherical surface. What about maps drawn on a surface like a donut – a surface
more interesting to mathematicians for its shape than its taste. For this surface,
Heawood showed that seven colours were both necessary and sufficient to colour
any map drawn on it. He even proved a result for a multi-holed donut (with a
number, h, holes) in which he counted the number of colours that guaranteed
any map could be coloured – though he had not proved these were the minimum
number of colours. A table for the first few values of Heawood’s h is: