4 Colour Theorem

Four Colours, A World of Possibilities

First posited in 1852, the Four Colour Theorem (or The Four Colour Conjecture) states that “no more than four colours are required to colour the regions of a map so that no two adjacent regions have the same colour.”

The theorem was first mentioned in a letter by Augustus De Morgan (1806 – 1871) having been brought to him by his student Frederick Guthrie who asked ‘what is the least possible number of colors needed to fill in any map (real or invented) so that neighboring counties are always colored differently?’

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  • The Five Colour Theorem was proven in the late 19th Century with relative ease but it took until 1976, and the aid of a computer, to crack the Four Colour Theorem (they say it was the first major theorem to be proved using a computer):

    ”If the four color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed that such a minimal counterexample cannot exist, through the use of two technical concepts .”

     four color tile map

  • However, the use of a computer to prove the theorem provoked immediate consternation from other mathematicians and philosophers who thought that, unless checked by humans, it was not ‘proved’ – the debate continues to this day, as there is no accepted solution to the theorem that doesn’t use a computer.

    Interestingly, whilst being the subject of much mathematical debate over the years, this theorem seems to have been little interest to cartographers who have many other issues to consider when colouring their maps!

    The proof and counterexample to this theorem is fabulously complicated to anyone other than mathematics whizz. Should you wish to delve deeper, watch this lecture by Robin Wilson (who wrote an entire book on the subject, Four Colors Suffice: How the Map Problem Was Solved) or this rather wonderful University of Cambridge educational page that explains it far better than we could!


    Fun for Kids!

    Have a look here to print out some fun colouring exercises for kids:
    Try colouring these map images with 4 colours so that the neighbours don't have the same colour.


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