The Seven Bridges of Königsberg

In 1736, Leonhard Euler solved the Königsberg bridge problem: the city had seven bridges connecting two islands in the Pregel river, and citizens wondered if one could cross every bridge exactly once and return to the start. Euler proved it was impossible — and in doing so invented graph theory and topology. His key insight: the possibility depends only on the number of bridges at each landmass, not on distances or geography. A physical puzzle became pure mathematics in a single step.