
Re: 4 colors problem
Posted:
Mar 7, 2014 1:04 AM


That's a very good point, thank you. The theorem does not guarantee a pattern or an algorithm. Only that there are 4 colors that exist to fill the map. Then what are the approaches or methods that are used to actually prove the theorem via supercomputers and mathematical equations? How do they do it? Only examples I've seen so far seem to center around variations of,
d d d d d d d d d d d d d a b a b a b a b a b a b b c d c d c d c d c d c d c
or...
a b c d c d a b
or...
a b a b c d c d
Patterns are not there in vain, actually, patterns do fulfill the promise of the theorem in certain limited situations. That's what makes them useful... to a point. The patterns discovered so far do a good job of algorithmically proving the theorem without covering all general, possible cases. Which gives one a (false) sense of hope for more general patterns.
When you do,
a b a b a b... c d c d c d... a b a b a b... c d c d c d...
there's no denying that the pattern complies with the 4 color theorem. This pattern is more compliant than the previous 4colorsperrow pattern because this one allows infinite horizontal expansion of a country on any row to any size to infinity. That solved the absurdity of the prior situation of 50% success/fail situation. It's not about pattern, yet one cannot but see some semblance of pattern at least in some portion of a map. For example, when a country is large enough that there are rows of smaller countries underneath it, then those small countries cannot take on the same color as the bigger country, they inevitably follow a checkered pattern using the remaining available colors.
As a simple example, if a row is repeated as
a b a b a b a b a b...
and the 2nd and 3rd countries unite, then
a b b "b" a b a b a b...
The quoted "b" changes to "a", which affects the rest of the row like,
a b b a b a b a b a...
And if that row happens to have infinite number of countries on it, then such a swap would go on infinitely, but at least it would come up with the correct answer for that row...
I hope some of all of this convey what I was hoping to say. It might take a very long time to prove or disprove it. It's taken 100+ years to prove it to the satisfaction of some if not all. It's taken equally as long to prove that it cannot be disproven. It might take 100+ years to prove it to the satisfaction of all, no doubting mind left unsatisfied. Or just as long to prove that it can be disproven in some minute cases.

