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Topic: 4 colors problem
Replies: 86   Last Post: Mar 13, 2014 4:36 PM

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 stumblin' in Posts: 58 Registered: 3/3/14
Re: 4 colors problem
Posted: Mar 5, 2014 4:11 PM

Partial coloring using a specific sequence of given 4 colors, followed by another specific (but different from the previous) sequence of given 4 colors (to be used for the next row of countries), known as a unit pattern, can follow simple rules to color the whole map (of the countries shaped as squares of equal size) guaranteeing that no matter how large the map gets (to infinity in all four directions), there will be no two countries sharing the same color, thus proving that the theorem is valid.

Given a=blue, b=green, c=red, d=yellow
The unit pattern is

a b c d
c d a b

This is like looking at a map from above and seeing 8 countries colored in colors denoted by a(=blue), b(=green), c(=red), d(=yellow).
It's the same as saying,

Country1 blue, Country2 green, Country3 red, Country4 yellow
Country5 red,Country6 yellow, Country7 blue, Country8 green

Repeat the pattern to the right to color more countries to the rightside, as needed.

a b c d a b c
c d a b c d

Repeat the pattern in right-to-left fashion to color countries to the leftside, as needed.

c d a b c d
d a b c d a b

Color rows of countries either above or below the unit pattern

c d a b
a b c d <-- unit pattern
c d a b <-- unit pattern
a b c d
c d a b

Counterexample is one instance that follows the theory's premise but does not reach the same conclusion as the theory promises.

Changing to above coloring to show one country twice bigger than the rest of the countries on the map

c d a b
a b c d
c d d b <-- two d's show one country twice big as the rest
a b c d
c d a b

Convention says common points don't count, only borderlines count, but here common points count the same as borderlines.

The above becomes,

c d a d <-- swapped d and b
a b c b <-- swapped d and b
c d d b
a b c b <-- swapped d and b
c d a d <-- swapped d and b

The 3 b's on the 4th row indicates a fail.

Date Subject Author
3/3/14 stumblin' in
3/3/14 Brian Q. Hutchings
3/3/14 stumblin' in
3/4/14 g.resta@iit.cnr.it
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3/4/14 g.resta@iit.cnr.it
3/4/14 magidin@math.berkeley.edu
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3/4/14 Brian Q. Hutchings
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3/4/14 Brian Q. Hutchings
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