11-13-2021, 09:13 PM
If the tools can't be applied to the problem, even after modifying them or the problem itself, then a different field is needed for sure. But all the fields of maths are interconnected that it's almost impossible to draw a nice defined border around a subject. As for the second part of your question, a famous example would be the Seven Bridges of Konigsberg. It required the invention of an entirely new field, graph theory.