You suddenly see how to solve the math problem that you have been working on for 30 minutes. In other words, you have used to solve the problem.

Insight

Explanation:

Insight is the ability to know and have a deep understanding of an area of study. It could also be about a person or a thing. Having insight means you are using your sixth sense or intuition to find a solution to a problem. For instance, you could gain an insight on how to fix a broken pipe, in a subject area in school, about a particular book, etc. Given the above, the math problem was solved as a result of insight.

answer; the automatic system because it is a "gut reactor"  

answer; preconventional;