There must be an in-between case where a small problem is solved by remembering the method. The kind of thing where you end up looking up the double-angle formulae or checking the definition for integration by parts. I guess the programming equivalent is remembering how to insert into a tree or do a bubble sort.

Problem-solving in the general case encompasses all of human creativity: I wouldn't claim that one becomes capable of tackling difficult problems through memorising techniques. But once you're part-way to solving a problem there is often a sequence of simpler steps (analysis, not synthesis) - surely most of us would like to be faster/more efficient/make fewer errors in these steps.