I'm pretty sure almost all mathematicians have been in a situation where they found an interesting problem; they thought of many different ideas to tackle the problem, but in all of these ideas, there was something missing—either the "middle" part of the argument or the "end" part of the argument. They were stuck and couldn't figure out what to do.
- In such a situation what do you do?
- Is the reason for the "missing part" the incompleteness in the theory of the topic that the problem is related to? What can be done to find the "missing part"?
For tenure-track/tenure professors, maybe this is not a big deal because they have "enough" time and can let the problem "stew" in the "back-burner" of their mind, but for limited time positions, for example, PhD, postdoc, etc., where the student/employee has to prove their capability to do "independent" research, so that they can be hired for their next position. I think for these people it is quite a bit of a problem, because they can't really afford to spend a "lot" of time thinking about the same problem.
I'm particularly interested in the answer in the realms of mathematics, but I would like hear suggestions from "non-mathematicians" as well.