The four colour conjecture
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If you have a map of any part of the world, or even one of an imaginary world, what is the smallest number of colours you can use in order to ensure that no two adjacent “territories” will have the same colour?
The answer is four. You can’t do it with three colours, because sooner or later two areas that are next to each other are bound to have the same colour. However, if you allow yourself four colours, it doesn’t matter how intricate your map is, you will always be able to colour it in such a way that any confusion can be avoided.
The next question is, how can you prove it? That is a very different matter, and it took until 1976 for an acceptable proof to be produced. This was done by mathematicians at Urbana, Illinois, who used thousands of possible scenarios and computer modelling to get to their solution. The final proof was nearly 1,000 pages long.
It sounds as though you ought to be able to imagine a map in which only four colours are used but in which there are adjacent territories that have the same colour, but just you try it! It doesn’t matter how convoluted your shapes are, or whether you have several large areas that adjoins dozens of tiny ones, you will never need to be in a situation where the rule is broken.
If you look at the United States map in the illustration, you will see that most of the map only needs three colours, with the crimson being needed on relatively few occasions – and it would also have been possible to use for orange for Wyoming and blue for Maryland. However, you can’t use “three and a half” colours so it has to be four!
Of course, if you want to keep your kids quiet on a wet Saturday afternoon, you could always get them to prove the rule the hard way!
The answer is four. You can’t do it with three colours, because sooner or later two areas that are next to each other are bound to have the same colour. However, if you allow yourself four colours, it doesn’t matter how intricate your map is, you will always be able to colour it in such a way that any confusion can be avoided.
The next question is, how can you prove it? That is a very different matter, and it took until 1976 for an acceptable proof to be produced. This was done by mathematicians at Urbana, Illinois, who used thousands of possible scenarios and computer modelling to get to their solution. The final proof was nearly 1,000 pages long.
It sounds as though you ought to be able to imagine a map in which only four colours are used but in which there are adjacent territories that have the same colour, but just you try it! It doesn’t matter how convoluted your shapes are, or whether you have several large areas that adjoins dozens of tiny ones, you will never need to be in a situation where the rule is broken.
If you look at the United States map in the illustration, you will see that most of the map only needs three colours, with the crimson being needed on relatively few occasions – and it would also have been possible to use for orange for Wyoming and blue for Maryland. However, you can’t use “three and a half” colours so it has to be four!
Of course, if you want to keep your kids quiet on a wet Saturday afternoon, you could always get them to prove the rule the hard way!
soncee › Thanks for sharing
fortune › I have never paid attention to colors on maps.
carmen3521 › ???
