“The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, and then to reduce some known NP-complete problem to it”
As a second example, in combinatorial game theory, the Sprague–Grundy theorem states
“every impartial game under the normal play convention is equivalent to a one-heap game of nim, or to an infinite generalization of nim. It can therefore be represented as a natural number, the size of the heap in its equivalent game of nim, as an ordinal number in the infinite generalization”
(https://en.wikipedia.org/wiki/Sprague–Grundy_theorem)
That means that, presented with an impartial game (both players can make the same moves, so chess is ruled out because white can’t move black pieces and vice versa) under the normal play convention (last player who can make a move wins), mathematicians look for a way to translate game positions to nimbers (https://en.wikipedia.org/wiki/Nimber) in order to learn how to play them.
Doing such mappings, if not trivial, requires creativity, so I think research on the subject would be in the psychology department.
Polya, in “How to Solve it” has some discussion on this (https://en.wikipedia.org/wiki/How_to_Solve_It#Heuristics)