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The Four Colour Theorem
Mathematicians and map makers share a lot of common ground. No more so than in the area of the four colour theorem.
This theorem simply states that any map in a single plane can be coloured using
four-colours in such a way that any regions sharing a common boundary (other than a single point) do not share the same
colour.
The theorem was first propounded by F Guthrie in 1853. Fallacious proofs have come and gone starting with Kempe in 1879 and Tait an 1880. In 1977 K. Appel and W. Haken used computer assistance to test many different combinations to effectively prove that four colours was all that was required in all instances. Since then, it may be that a mathematical proof has, at last, been arrived at.
So, if we know that we can colour any map using just four colours how to we go about it. A little though would indicate that the problem is not straightforward. Simply starting with a random colour and an arbitrary polygon would soon lead to an impasse when the process met an area bounded by more than three other areas yet to be
coloured.
Kempe is credited with first recording that, when tackling a map of national boundaries, those with three or fewer neighbours presented no problems. His solution was to ignore (temporarily remove from the map) those countries with three or fewer
neighbours. This process will immediately simplify the remainder of the map and can be repeated until only countries with three or fewer neighbours remain. The remaining areas can then be
coloured. Then the missing countries can be restored in reverse order to their removal and coloured as the process proceeds. This effective procedure is know as Kempe transformations.
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