A good programmer working intensively on his own code can hold it in his mind the way a mathematician holds a problem he's working on. Mathematicians don't answer questions by working them out on paper the way schoolchildren are taught to. They do more in their heads: they try to understand a problem space well enough that they can walk around it the way you can walk around the memory of the house you grew up in. At its best programming is the same. You hold the whole program in your head, and you can manipulate it at will.

Your code is your understanding of the problem you're exploring. So it's only when you have your code in your head that you really understand the problem.