[genewitch]

there has to be a layman's explanation. I knew the easiest way to get four colors was to put a split square inside another split square "donut", but the reason you can't force 5 colors is that word "inside". There has to be a nice, tidy "verbal proof" that no matter what, one or more colors will be "trapped" inside at most 3 other colors.

[waterproof]

You would think. The easier, five-color theorem proof fits in a few paragraphs but the four-color theorem really resists a simple explanation. Even the simplified 1990s version of the proof (which came ~20 years after the original proof and 100 years after the 5CT proof) required enumeration of hundreds of individual cases.

https://en.m.wikipedia.org/wiki/Four_color_theorem