Each of these phases has its importance. It may happen that a student hits upon an exceptionally bright idea and jumping all preparations blurts out with the solution. Such lucky ideas, of course, are most desirable, but something very undesirable and unfortunate may result if the student leaves out any of the four phases without having a good idea. The worst may happen if the student embarks upon computations or constructions without having understood the problem. It is generally useless to carry out details without having seen the main connection, or having made a sort of plan. Many mistakes can be avoided if, carrying out his plan, the student checks each step. Some of the best effects maybe lost if the student fails to reexamine and to reconsider the completed solution.