Four Colour Problem

[Diogenes]

It's easy to show that five regions on a plane cannot be arranged so that they all touch each other, but apparently this doesn't constitute proof of the four colour theorem. Why not?

[rja211077]

picture a pie chart divided into x segments - surely that is x regions on a plane all touching each other (at the centre point of the circle)?

[Quizmonster]

Click http://bhs.broo.k12.wv.us/discrete/4Color.htm for a relevant website. I'm no mathematician, but it may help you.

[WaldoMcFroog]

From a mathmatical point of view, and struggling to remember what it said in Simon Singh's excellent 'Fermat's Last Theorem' I think I know why.

A proof is mathematical formula that shows definintively that what one is trying to show is absolutely correct for all possible values. Although one can demonstrate that five colours cannot be arranged on a plain such that they all touch, this doesn't - of itself - prove that the same is true for six colours. Although one can definitely demonstrate that six colours cannot be arranged on a plain such that they all touch, this doesn't - of itself - prove that the same is true for seven colours, and so on...

To constitute a mathmatical proof, the formula must prove that any number of colours over three (I presume) cannot be placed on a single plain such that they touch. God knows what the proof would be though!

[johnah]

It may be easy to draw sets of four mutually touching areas that it is not possible to add a fifth area to which would touch all four. However you cannot be sure that you have examined all possible sets. That is why it cannot constitute a proof.