Seven Bridges of Königsberg

A city had seven bridges.

People wanted to walk them all. They wanted to cross each bridge just once. A man named Euler tried to find a way.

He found it was not possible. He saw that the shape of the land did not matter. Only the number of bridges mattered. One land had five bridges. Three lands had three bridges. These are odd numbers. Because of this, the walk could not work. Euler's work helped start a new kind of math.

Long ago, a city called Königsberg had a puzzle. The city sat on a river. There were two large islands in the water. Seven bridges connected the islands to each other and the mainland.

People wanted to walk a path through the city. The goal was to cross every bridge exactly one time. A man named Leonhard Euler looked at this problem in 1736.

He found that no such path exists.

Euler realized the exact shape of the land did not matter. He only needed to know how the parts were connected. He turned the land into points called vertices. He turned the bridges into lines called edges. This new way of looking at things is called graph theory.

Euler saw that most points must have an even number of edges. This is because you enter a point and then leave it. In Königsberg, all four land masses had an odd number of bridges. One had five bridges. The other three had three bridges each. Because of these odd numbers, the walk was impossible. His work helped start a new field called topology. This math studies how things are connected rather than how big they are.

Long ago, a city called Königsberg sat by the Pregel River in Prussia.

The river divided the land into two main parts. There were also two large islands named Kneiphof and Lomse. Seven bridges connected all these pieces of land together. People loved to wonder if they could walk a special path through the city. This path would have to cross every single bridge exactly one time. They could not use any other way to get to an island. They also could not cross a bridge without going to the other side.

In 1736, a mathematician named Leonhard Euler solved this puzzle. He realized that the exact shape of the land did not matter. He did not need to know the size of the islands. He only needed to know how the land masses were connected. Euler turned the land into points called vertices. He turned the bridges into lines called edges. This new way of thinking is called graph theory. By using this method, he could study the connections instead of the distances. He showed that the path people wanted was actually impossible to walk.

Euler looked closely at how many bridges touched each piece of land. He called this the degree of a vertex. He noticed a pattern for any successful walk through a graph. If you enter a piece of land, you must also leave it. This means most land masses must have an even number of bridges. An even number means they can be paired up for entering and leaving. Only the very start and the very end of a walk can be different. These two points can have an odd number of bridges.

In the city of Königsberg, the numbers did not work out. One land mass had five bridges touching it. The other three land masses each had three bridges. Since all four pieces of land had an odd number of bridges, the walk could not happen. A successful walk can only have zero or two land masses with odd numbers. Euler presented his ideas to the St. Petersburg Academy in 1735. He published his full solution in a journal in 1741. His work was a very important start for the study of networks.

This discovery changed how we think about the world of math. It helped create a field called topology. Topology studies how things are connected rather than their exact size or shape. This is different from traditional geometry that uses measurements. Today, you can see models of these bridges in many places. Some universities have built models in their grass or on their sidewalks. Even in cities like Bristol, people still enjoy bridge walking puzzles. Euler's ideas still help us understand how many different things connect today.

The Seven Bridges of Königsberg is a famous problem in the history of mathematics. It began as a puzzle in the Prussian city of Königsberg, which is now Kaliningrad, Russia.

The Pregel River flowed through the city and divided the land. There were two large islands, Kneiphof and Lomse, and two mainland areas, Altstadt and Vorstadt. Seven bridges connected these four land masses to one another. The challenge was to find a walking route that crossed every bridge exactly once. To solve the puzzle, a person could not reach an island by any means other than a bridge. They also could not cross a bridge without traveling all the way to the other side.

In 1736, the mathematician Leonhard Euler provided a negative resolution to this problem. He proved that such a walk was actually impossible. To do this, Euler developed a new technique of analysis. He realized that the specific layout of the city was not the most important part. The exact shape of the islands and the length of the bridges did not matter. Instead, the only thing that mattered was the sequence of bridges crossed. This insight allowed him to simplify the problem into abstract terms.

Euler's method involved creating a mathematical structure called a graph. In a graph, each land mass is replaced by an abstract point called a vertex or a node. Each bridge is replaced by an abstract connection called an edge. The edge only serves to show which pair of vertices is connected. Because only the connections matter, the shape of a graph can be distorted without changing its meaning. It does not matter if the edges are straight or curved. It also does not matter where one node is placed relative to another.

Euler analyzed how many bridges touched each land mass. He called this number the degree of a node. He observed a specific rule for any successful walk through a graph. Except for the very start and the very end, every time you enter a vertex via a bridge, you must leave it via another bridge. This means that for any middle point in a walk, the number of bridges must be even. One bridge is used to enter, and another is used to leave. If you visit a node multiple times, you need an even number of bridges to account for every entry and exit.

In the original Königsberg layout, the numbers did not allow for this pattern. One land mass was touched by five bridges, while the other three were touched by three bridges each. Because all four land masses had an odd degree, a single continuous walk was impossible. Euler showed that a walk traversing every edge exactly once requires the graph to be connected and have exactly zero or two nodes of odd degree. If there are two odd nodes, the walk must start at one and end at the other. This type of walk is known as an Eulerian trail or an Euler walk.

Euler presented his work to the St. Petersburg Academy on August 26, 1735. He published his findings in 1741 in the journal Commentarii academiae scientiarum Petropolitanae. His work is now considered the first theorem of graph theory. It also served as the first true proof in network theory, which is a branch of combinatorics. His ability to look past measurements to see structural connections foreshadowed the development of topology. Topology is a field that studies properties that remain unchanged even when an object is stretched or distorted.

Today, the bridges of Königsberg have changed significantly. Some were destroyed during the bombing of the city in World War II. Others were demolished to make room for highways. Currently, five bridges exist at the original sites. This new configuration changes the math of the city. Two nodes now have a degree of 2, and two have a degree of 3. Because there are now only two nodes with an odd degree, an Eulerian path is actually possible. This shows how changing the physical connections of a system changes its mathematical properties.