On Tuesday, March 4, 2014 5:18:25 AM UTC+1, stumblin' in wrote: > I have (what I hope is) a simple line of thought that shows why 4 color solution works only half the time in some situations.
From a mathematical point of view "some situations" is not very precise, nor particularly relevant, imho.
In your example, you start with 4 colors, but a checkered map can be colored with just two colors.
Actually it is not clear what do you mean with "4 color solution only works half the time", since we already know that 4 colors are ALWAYS sufficient.
Clearly, if your map changes and you are not allowed to change colors already assigned, then you can construct maps which need an arbitrary number of colors, even if 2 colors are sufficient if the last map can be colored from scratch.