4 color theorem.

I am going to take again an opportunity to make a fool of myself and to let more people know about it by using english.
Let's do it:
So I have first understood that there was some difficulties to know if 4 colors were enough to color a chart of countries. It was so difficult that they needed a computer to do it. I don't know which method they have used.
But I decided instead to ask myself if it was possible to draw 5 countries having a common border with each other.
And of course I failed as would have failed anybody wanting to verify if they would need a fifth color to color a chart.
But I decided to struggle because if I would find something with only five areas it might be possible to make it true for all the plan and perhaps later for the space.

So I started with one area and added a second one, a third one and when I wanted to add the fifth one I didn't have any choice than going througth an existing area. Because, when reaching level 4, there is always an area that is completely enclosed by the 3 other ones.

At this point a lot of problems have to be resolved:
. For exemple what if the common border between two or more areas is only a point? Without having get a valid demonstration I let myself think that I could accept that two or more areas in this situation can be considered as one area (they can use the same color). It is as if an area hade been compressed in the middle until two opposed frontiers would join just by one point.
. Should we considere the form of each area as having some importance?
Without finding it a real demonstration I accepted that it had no importance. I mean you can change the shape of any area without changing the problem (as moving a thread that is the border). The border has only to keep more than one point.
. What if there is an area without color? I would considere no color as a color by itself.
. There is also a problem with the starting configuration. How can I be shure I have not missed a configuration that would permit to do it with 5 areas? This is one of the biggest point I think. The solution might be to reduce all case to one or a few simple cases. But how?
. What if it was possible to do it with n>5 areas even if it's impossible with 5?
Again I would considere a kind of recursive demonstration showing that if it is true at level 5 it will be true also at level 6 because we can isolate the 5 areas among the 6 ones and so on.

Finally some mathematical equation made with the help of the set theorie might give me the reply. But I don't have the knowledge (I can't ask my wife, I am single!).
If your are interested you can always try with your pen.


Suppose somebody would find a satisfying answer and a simple demonstration for this first step (finding that it is impossible to put five area having a common border with every other one, the border between two areas being not a single point) then I still have to find a demonstration for the entire plan.

I have some idee again but, is it a demonstration?
Suppose we have a plan with n areas. I could start building separately a copy of this plan starting with area number one and add one area at a time. Normally using this method I shouldn't need modifying any area already drawn. At each step I would find myself in the same situation as the one described in the first part of this study (the impossibility to draw 5 countries having common border).
The point is that if it is possible to draw a plan with some countries having (in a group of 5) common border with each other then -when we are building it- we would encounter the basic situation described in the first part. And we know we couldn't because it is impossible to do it. So such a plan wouldn't exist and we only need 4 colors.

But again I don't think this is a satisfying demonstration.Every basic elements of the demonstration I have in mind are written in this text but I will have to rewrite it in a different order to make it more obvious (even for me).

It is likely I am breaking an open door or I am not breaking anything at all.
Do you find it interesting?