The 18th-century Prussian city of Königsberg sat across a river split into two channels around an island, and seven bridges connected its four landmasses to each other. A local puzzle asked whether you could walk a single route that crossed every one of the seven bridges exactly once and returned you to where you started. People had tried and failed for years by the time Leonhard Euler heard about it in 1735. He didn't solve it by finding a route no one else had thought of. He solved it by proving no such route could exist — and the method he used to prove that is now considered the starting point of graph theory.
Throw away everything except the connections
Euler's move was to strip the map down to only what actually mattered for the question being asked. The exact shape of the island, the width of the river, the distance between bridges — none of it affects whether a round-trip crossing every bridge once is possible. What matters is only which landmasses connect to which, and how many bridges connect each pair. So he reduced the whole city to four points, one per landmass, joined by seven lines, one per bridge — the object mathematicians now call a graph, with the landmasses as vertices and the bridges as edges. Removing every detail that didn't bear on the question turned an intractable walking puzzle into a shape simple enough to reason about directly.
Once the city was a graph, Euler noticed something about what a bridge-crossing walk requires at each landmass along the way. Every time the walk passes through a landmass — arriving over one bridge, leaving over another — it uses up two of that landmass's bridges, one in and one out. A landmass you only start or end at is the exception, needing just one bridge used at that point. For a walk to cross every bridge exactly once and still get through each landmass cleanly, every landmass's bridges have to pair off evenly — which means every vertex needs an even number of edges, except possibly the start and end points if they're different places. Königsberg's four landmasses all had an odd number of bridges. Not most of them — all four. That single fact, and nothing about the specific geography, is what makes the walk impossible.
A path, not just a loop
The condition changes slightly if the walk doesn't need to return to its start: a route that crosses every edge exactly once but ends somewhere different is possible if exactly two vertices have an odd number of edges — the start and the end — with every other vertex still even. Königsberg fails even this relaxed version, since it has four odd vertices, not two. Either way you ask the question, the answer for that particular map is no, and Euler's paper on it, published in 1736, is generally credited as the first paper in graph theory as its own subject rather than a tool borrowed from geometry.
What we're still unsure about
None of this makes Königsberg's bridges special — Euler's point was closer to the opposite, that the specific city stopped mattering the moment the graph was drawn. Historically, the puzzle is also less permanent than it sounds: wartime bombing and later reconstruction changed the city, now Kaliningrad, so the modern bridge layout isn't the same seven bridges Euler analysed, and reproducing the exact 1735 puzzle on the ground today isn't straightforwardly possible. The mathematics survived the city that inspired it; the city itself didn't survive unchanged.
This sits inside Graph Theory, one of eight topics in Discrete Mathematics, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.