Oh I think I found the solution. This time everything leads to right answer. There is a unit pattern more fundamental than the original one. Instead of, a b c d c d a b
Use, a b c d
Follow the same rule for coloring more countries a b a b a b c d c d c d
b a b c d c d
c d a b <-- unit pattern c d <-- unit pattern a b c d
This works flawlessly even with a country whose horizontal length is bigger than others on the map. In the process of disproving a theorem I became a complete believer. The theorem was proposed during the time when there was no computer. The (modified and simplified) theorem is proved without using a computer.