1. Introduction "A sudden inspiration is often the result of a long period of hard work." – Thomas Edison A central but often overlooked feature of problem-solving is that it does not begin with clarity. Contrary to the polished presentation of solutions in textbooks and formal instruction, real thinking typically starts in a state of vagueness. The problem is only partially understood, relationships are indistinct, and ideas exist as fragments rather than structured arguments. This condition, what may be called blurry thinking , is not a defect of reasoning but its natural starting point (Hadamard, 1945). The work of mathematician Jacques Hadamard provides one of the most influential accounts of how such indistinct thinking becomes precise. Building on insights from Henri Poincaré [1] and Graham Wallas (an English sociologist), Hadamard described problem solving as a staged progression through preparation, incubation, illumination, and verification (§3-§6). This frame...
Random thoughts and used ideas