Regardless of your research area or career stage, you almost certainly have experienced "mistakes" or "failures" in your research. This colloquium talk will introduce three crucial developments in number theory, alongside some of the mistakes, false starts, and otherwise suboptimal approaches that led to them. These three strands--one from each of the 18th, 19th, and 20th centuries--eventually merged spectacularly to enable progress in the 21st century, including in the speaker's work. Intertwined with these specific mathematical triumphs are valuable lessons about grappling with -- and sometimes adding to -- confusion while trying to solve hard problems. The focus of this talk will be on progression of key ideas, rather than on technical details. Graduate students and faculty from all areas of math are encouraged to attend.