Henry Ernest Dudeney/Modern Puzzles/140 - The Four-Colour Map Theorem/Refutation

== Refutation of by : $140$ - The Four-Colour Map Theorem ==


 * The Four-Colour Map Theorem

Refutation
correctly shows that no more than $4$ regions can be drawn so that each borders all the others.

However, his proof fails to show that four colors are sufficient for all maps.

It is indeed true that if any four regions of a map are considered in isolation, then no fifth color is necessary for any fifth region.

What is required, however, is a proof that on a map with a large number of regions, these various sets of five will never conflict with each other in a way that five colors are demanded.

The difficult is best seen by actually constructing a complicated map, using the step by step procedure proposed by.

Suppose that each new region is drawn so as to border three others.

Then its color is determined automatically, and four-color maps can indeed be extended indefinitely.

However, suppose that many new regions are added that touch either one, two, or even no previously drawn regions.

Then the choice of colors for these regions suddenly becomes arbitrary.

As the map grows in size and complexity, it is suddenly discovered that it is possible to add a new region that will require a fifth color.

By backtracking and altering previous decisions as to what color to be used, it appears that it is always possible to correct the mistake and accommodate the new region without needing that fifth color.

But using this process it cannot be certain that it is always possible.