Seven Bridges of Königsberg

The Seven Bridges of Königsberg is the problem that launched both graph theory and topology. In 1736, the Swiss mathematician Leonhard Euler was asked whether it was possible to walk through the Prussian city of Königsberg, crossing each of its seven bridges exactly once, and return to the starting point. Euler proved that no such walk existed — not by enumerating every possible route, but by abstracting the city map into a graph of land masses (nodes) and bridges (edges), then proving that such a circuit requires every node to have an even number of edges.

The proof was revolutionary because it showed that the answer depended not on the geometry of the city — the lengths of bridges, the sizes of islands — but purely on how the pieces were connected. This abstraction of structure from measure is the foundational move of topology and the foundational insight of network analysis. Every problem in network routing, circuit design, and logistical optimization is a descendant of Euler's 1736 paper. The bridges themselves were destroyed in World War II, but the problem is immortal.