4 colour theorem

Birmingham Wards coloured in using 4 colours. There might be a more pleasing distribution of the colours.

“A student of mine [Guthrie] asked me today to give him a reason for a fact which I did not know was a fact – and do not yet. He says that if a figure be anyhow 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 following is the case in which four colours are wanted. Query cannot a necessity for five or more be invented…” Quoted from a letter by Augustus DeMorgan after Francis Guthrie asked about the number of colours needed.

The four colour theorem was the first major theorem to be solved by relying on a computer algorithm (with the provisio that the theorem depends on the compiler not having any logical bugs).

Using one of Eric Harshbarger’s Java applets , I have hacked up a page where you can colour in a map of the new Birmingham City Wards. Actually walking through the colouring of this small map makes you think through aspects of the problem. Quite often, you have to change one of the colours to avoid having the same colours on a boundary Рraising the issue of how many logically distinct colourings can their be for each (finite) map?

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