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...
1. Introduction The capacity to solve problems is frequently linked to intelligence, especially as quantified by IQ scores. However, while cognitive ability confers certain advantages, such as faster pattern recognition or stronger working memory, it is not the primary determinant of effective problem-solving. High IQ may accelerate initial understanding, yet it cannot replace accumulated experience or the disciplined habits required for sustained success. Instead, the systematic study of problems across diverse disciplines and difficulty levels emerges as the more dependable pathway to genuine problem-solving proficiency. This essay examines how repeated engagement with problems cultivates essential habits, transferable analytical skills, intellectual humility, productive intuition, and discipline, ultimately enabling individuals to navigate novel challenges with confidence. Basically, to solve problems, you must cultivate experience and a love for...