I agree, and it would be amazing for someone to find a deeper intuitive reason for why 15 is the answer to this problem, or 8 is the answer to that other proble

I agree, and it would be amazing for someone to find a deeper intuitive reason for why 15 is the answer to this problem, or 8 is the answer to that other problem, or many such cases. This is however extremely hard in my opinion (note that I say "opinion" on purpose, because I don't have any way to formalize this idea). Let me use a couple more examples: any Rubik's cube (3x3x3) can be solved in at most 20 moves (on the standard metric, counting double turns as one move), and a Sudoku needs at least 17 clues to have a unique solution. Why 20? Why 17? We (humanity) got to both answers through computation, and even though the computation requires mathematical observations to be made more efficient, at the end we still checked an in-human number of cases... So in all honesty, if I had to bet, I'd say we will probably never intuitively know why 20 moves or why 17 clues, or in my case, why 15 colors... I'd love to be proven wrong tho!