Introduction
Growing up I read a number of short articles about The Seven Bridges Of Königsberg written for children. I was intrigued and already hooked on math! I wanted to know more, but at the time not much was written on the problem and not much could be found in the local library. Now I can look even deeper into the problem with the information gathered below. Even if you are not interested in graph theory but are interested in mathematics, I suggest you take a look into a most intriguing problem.
Problem
Way back in 1735, Euler heard about an interesting problem that the town of Königsberg was facing.
At the time, Königsberg was a city in Germany, and the city was built around a river called the Pregel River. This city thrived with its merchant economy and part of the reason that it did so well was because it was structured in a particularly interesting way. There were two large islands in the middle of the Pregel River, and they were each connected to one another, as well as to the two riverbanks on either side, which comprised the majority of the city. And how were they connected? By bridges, of course! Seven of them, in fact.
The citizens of Königsberg spent their Sundays walking around town, enjoying their beautiful city. In the process, they came up with a game — which, as it turned out, proved to be incredibly difficult to accomplish. The goal was to walk across all of the seven bridges crossing the islands only once, without ever repeating a single bridge in the course of one’s walk.
At first, when people asked Euler to solve this problem, he brushed it off, thinking that it had nothing to do with mathematics, and therefore wasn’t really worth his time. But the more that he thought about it, the more intrigued he became. In a letter to a mathematician friend of his, he wrote:
This question is so banal, but seemed to me worthy of attention in that geometry, nor algebra, nor even the art of counting was sufficient to solve it. In view of this, it occurred to me to wonder whether it belonged to the geometry of position which Leibniz had once so much longed for. And so, after some deliberation, I obtained a simple, yet completely established, rule with whose help one can immediately decide for all examples of this kind, with any number of bridges in any arrangement, whether such a round trip is possible, or not….
Letter Euler wrote to the Italian mathematician
and astronomer Giovanni Jacopo Marioni
He was hooked. Euler was so entranced, in fact, that he ended up writing a paper later that year that would contain a solution to the bridge problem. But before we understand how Euler solved this problem, we need to cover a few basic graph theory rules first. [1]
References
[1] ⭐ Joshi, Vaidehi. “Königsberg: Seven Small Bridges, One Giant Graph Problem”. 2017. Medium. https://medium.com/basecs/k%C3%B6nigsberg-seven-small-bridges-one-giant-graph-problem-2275d1670a12.
Additional Reading
al Conocimiento, Ventana. “Euler And The Königsberg Bridge Problem | OpenMind´s Puzzles”. 2022. OpenMind. https://www.bbvaopenmind.com/en/science/mathematics/euler-konigsberg-bridge-problem/.
“Activity: The Seven Bridges Of Königsberg”. 2022. mathsisfun.com. https://www.mathsisfun.com/activity/seven-bridges-konigsberg.html.
“Eulerian Path – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Eulerian_path.
“Eulerian Path And Circuit For Undirected Graph – GeeksForGeeks”. 2013. GeeksForGeeks. https://www.geeksforgeeks.org/eulerian-path-and-circuit/.
Joshi, Vaidehi. “A Gentle Introduction To Graph Theory”. 2017. Medium. https://medium.com/basecs/a-gentle-introduction-to-graph-theory-77969829ead8.
Labh, Sunny. 2024. “How Königsberg’s 7 Bridges Problem Inspired the Inception of Graph Theory.” Medium. Cantor’s Paradise. https://www.cantorsparadise.com/how-k%C3%B6nigsbergs-7-bridges-problem-inspired-the-inception-of-graph-theory-6283cad7bf10.
Have you ever wondered how a simple stroll across a river could inspire a whole new branch of mathematics? That is exactly what happened in the 18th century, when a curious puzzle about the city of Königsberg sparked the birth of graph theory. The problem is based on an actual geographical setting in the city of Königsberg, which was in Prussia (now Kaliningrad, Russia). The city was divided into four regions by the Pregel River and the regions were connected by seven bridges. The challenge posed by this setup was this: Can you walk through the city in such a way that you cross each of these seven bridges exactly once?
McBride, Martin. 2023. “The Seven Bridges of Königsberg.” Medium. October 5. https://medium.com/recreational-maths/the-seven-bridges-of-k%C3%B6nigsberg-83494975c186.
The problem of the seven bridges of Königsberg was a popular problem in mathematics in the early 1700s. It was solved by Euler in 1736, and although the problem itself wasn’t especially difficult, Euler’s solution laid some of the groundwork for modern graph theory.
Merriam, Areeba. “Euler and the Seven Bridges of Königsberg”. 2023. Medium. https://www.cantorsparadise.com/euler-and-the-seven-bridges-of-k%C3%B6nigsberg-c8d80ce65094.
The ideas developed in solving the Seven Bridges of Königsberg problem were crucial for the development of the subject of graph theory and topology, which is a branch of mathematics concerned with the study of geometric objects that are insensitive to certain kinds of transformations, such as stretching or bending.
In particular, the concept of a graph provided a way to represent the connectivity of space, while the theory of the Eulerian path provided a way to understand the structure of the graph. These concepts were further developed and generalized to give rise to the modern field of topology.
Paoletti, Teo. “Leonard Euler’s Solution To The Konigsberg Bridge Problem | Mathematical Association Of America”. 2022. maa.org. https://www.maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem.
Patowary, Kaushik. “What Mathematics Has To Do With The Seven Bridges Of Königsberg”. 2022. amusingplanet.com. https://www.amusingplanet.com/2018/08/the-seven-bridges-of-konigsberg.html.
Today, graph theory is a major tool in mathematical research, and finds application in diverse fields such as electrical engineering, computer programming and networking, business administration, sociology, economics, marketing, and communications—the list goes on and on. Recent research suggests that graph theory can provide fascinating insights into the breakdown of communication between nerve cells that results in the progressive memory loss of Alzheimer’s.
Sethi, Rahul. “The Seven Bridges Of Königsberg”. 2020. Medium. https://medium.com/stamatics-iit-kanpur/the-seven-bridges-of-k%C3%B6nigsberg-268f73122e82.
“Seven Bridges Of Königsberg – Wikipedia”. 2015. En.Wikipedia.Org. https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg.
St. Clair, Natalya. “Graph Theory Problems, The Seven Bridges of Königsberg Problem”, 2015. mathcircle.berkeley.edu. https://mathcircle.berkeley.edu/sites/default/files/archivedocs/2015/lecture/Graph%20Theory%20Intermediate%20I%20and%20Ii-2.pdf.
Wilkins, James. 2025. “The Bridges of Königsberg.” Medium. Puzzle Sphere. https://medium.com/puzzle-sphere/the-bridges-of-k%C3%B6nigsberg-707db5b9ff4b.
The Bridges of Königsberg may have seemed like a whimsical puzzle to the residents of its time, but it has had far wider implications for the study of mathematics since then.
Euler’s solution to this problem is widely regarded as the birth of the field of graph theory, a very different mathematical discipline from the algebra and equations that came before it. Since then, it has led to the development of many different ideas within and beyond the world of mathematics. The internet, maps, navigation systems, and delivery networks all depend on ideas from Graph theory.
While the bridges may have changed, their legacy remains. It serves as a timeless reminder of how simple questions can lead to profound discoveries that reshape the way we understand the world.
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
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