*edit* Removing the fluff, the question is:

When solving problem X by heart, how does the mind reaches the solution very fast? by 'running' an algorithm or 'accessing' a table

The 'fluffy' version:

Since I was a kid I was always amazed by mental arithmetic. Today I still am.

With time I got to realize that the best engineers, physicists, mathematicians around me seem to not only handle very fast and precise mental arithmetic but also fast and precise mental proportions and spatial relationships.

therefore my question is: are they solving these problems in a algorithm like fashion (but faster than most people) or are they getting these results from the back of their mind from some sort of "table", where they archived many results through the years and just retrieve them "without effort"?