Königsberg, now Kaliningrad, was built on both banks of a river and on two islands in it, joined by seven bridges. A popular puzzle asked for a walk that crossed every bridge exactly once.
In 1736 Leonhard Euler showed there was no such walk, and he did it without trying routes. He threw away the map. Each piece of land became a point, each bridge a line between two points, and the question became one about points and lines.
A walk that uses every line once can only pass through a point by arriving on one line and leaving on another, so every point it passes through needs an even number of lines. Königsberg had four pieces of land, and all four had an odd number of bridges. No walk was possible.
1. Why It Mattered
Euler’s trick, keeping only what is connected to what, is now called graph theory. It is the maths behind route planning, the way a phone network finds a path, and the way a search engine ranks pages.
The lesson for a student is smaller and just as useful: when a problem is hard, ask what you can leave out. Euler left out the whole city.