I am convinced there has to be a way to leverage the expressive power of a language like pure prolog without the strain of the 'hints' you need to give it when

I am convinced there has to be a way to leverage the expressive power of a language like pure prolog without the strain of the 'hints' you need to give it when the depth-first algorithm can't solve the problem. My bet is that if the interpreter 'knows' enough algorithms to solve common calculations, it could leverage these to compute 90% of problems (without needing to have one algorithm for each problem). For the other 10% you would have to supply the interpreter with the necessary algorithms yourself.