One reason is that people who like to solve this do it for artistic reasons. They do it because they like to, they do it because the approach can be elegant and beautiful. In rare cases analytic solutions lead to generalizations that turn into theorems. I would say that most of mathematics have been created by people's own self-interest in solving problems that many would find "useless".

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"Democracy has proved only that the best way to gain power
over people is to assure the people that they are ruling
themselves. Once they believe that, they make wonderfully
submissive slaves." - Joseph Sobran