If we keep looking at a problem in one way, we may not come up with a solution. However, if we change the way we look at it a solution might click. Mind needs cues. It works on patterns. The new point of view may match an already existing pattern and you can come up with a solution.
Integer Multiplication: we know how to do it, right? Why not write a recursive algorithm for the same problem? It is not immediately obvious the benefit of this. The time complexity is the same. But the representation of the new algorithm i.e. a recurrence suggests a way you can improve it further, thus Guass’s multiplication formula.