“The famous French mathematician Henri Poincaré was very interested in mathematical creativity. He describes a period of hard and seemingly fruitless effort to solve a problem, from which he took a break to join a geological expedition. As he was stepping on a bus, he made one of the most important breakthroughs of his life. The solution came to him out of nowhere, and was accompanied by a perfect certainty as to its correctness. Poincaré did not claim that this was a miraculous incident. Indeed, he believed that we can solve problems when we are not consciously thinking about them.
At any moment we are aware of only a fraction of what goes on in our brain. Poincaré was quite aware that creativity requires a period of conscious effort followed by a period of rest. Our unconscious mind keeps working on the problem behind the curtain. As a consequence sometimes a solution, or at least a good idea, will emerge apparently out of nowhere. A period of concerted effort to check the idea and put it in a form that is understandable to others is then necessary. Poincaré's contemporary Albert Einstein may have expressed this most succinctly when he said that ‘Creativity is the residue of time wasted’.”