The sudden awakening

One phenomenon is certain and I can vouch for its absolute certainty: the sudden and immediate appearance of a solution at the very moment of sudden awakening. On being very abruptly awakened by an external noise, a solution long searched for appeared to me at once without the slightest instant of reflection on my part—the fact was remarkable enough to have struck me unforgettably—and in a quite different direction from any of those which I had previously tried to follow. Of course, such a phenomenon, which is fully certain in my own case, could be easily confused with a “mathematical dream,” from which it differs.

—Jacques Hadamard, An Essay on the Psychology of Invention in the Mathematical Field