Why the Seven Bridges Could Not Be Crossed Once Each
In 18th-century Königsberg, the Pregel River divided the city into land areas linked by seven bridges. The puzzle was not simply to find a clever route: it asked whether a walk could cross every bridge exactly once, regardless of where it began or ended.
Leonhard Euler’s key move was to discard everything that could not affect the answer. The shape of the river, the length of the bridges, and the streets between them did not matter. What mattered was which areas each bridge connected. He represented each land area as a vertex—a point—and each bridge as an edge, a line joining two points. This kind of simplified network is now called a graph.
The Odd-Number Test
Count the bridges touching each land area. In any route that uses every bridge once, an area entered along a bridge must also be left along another. Those crossings pair up. Only the route’s starting area and ending area can have an unpaired crossing, so at most two areas can have an odd number of bridges attached to them.
In the Königsberg network, all four land areas had odd counts. No route could satisfy the rule, so no amount of trial and error—or a different choice of starting point—could solve the puzzle.
A New Kind of Mathematical Question
Euler’s result mattered because it answered a question about possible connections without calculating distances or drawing an exact map. The same way of thinking can be used wherever relationships matter more than physical layout: in networks of roads, communications, or other linked systems. The memorable conclusion was stark: the bridges were not arranged badly for a particular walker; their pattern of connections made the requested walk impossible.
