Finding Königsberg on a modern map might be a challenge, but its unique geography has made it a legendary city in the world of mathematics. This medieval German city was situated on both sides of the Pregel River and featured two large islands at its center. These islands were connected to each other and the riverbanks by seven bridges.
Carl Gottlieb Ehler, a mathematician who later became a mayor, was fascinated by these islands and bridges. He pondered a seemingly simple question: Is there a route that allows someone to cross all seven bridges without crossing any of them more than once?
Take a moment to think about it. Stumped? It turns out it’s impossible. However, trying to explain why led the renowned mathematician Leonhard Euler to create a new branch of mathematics. Ehler reached out to Euler for assistance with the problem. Initially, Euler thought the question was unrelated to mathematics. But as he delved deeper, he realized there was something significant to uncover.
Euler’s solution was linked to a type of geometry that didn’t exist yet, which he called the Geometry of Position, now known as Graph Theory. His first insight was that the specific path taken between entering and leaving an island or riverbank didn’t matter. This allowed the map to be simplified, with each of the four landmasses represented as a single point, or node, and the bridges as lines, or edges, connecting them.
This simplified graph made it easy to count the degrees of each node, which is the number of bridges touching each landmass. Why are these degrees important? According to the rules, once travelers arrive at a landmass by one bridge, they must leave via a different bridge. This means the bridges leading to and from each node must occur in pairs, making the number of bridges touching each landmass even. The only exceptions are the starting and ending points of the walk.
Examining the graph, it becomes clear that all four nodes have an odd degree. Therefore, no matter the path chosen, a bridge will have to be crossed twice. Euler used this observation to develop a general theory applicable to all graphs with two or more nodes. An Eulerian path, which visits each edge only once, is possible in two scenarios. The first is when there are exactly two nodes with an odd degree, and all others are even. In this case, the path starts at one odd node and ends at the other. The second scenario is when all nodes have an even degree, allowing the path to start and end at the same location, forming an Eulerian circuit.
So, how could you create an Eulerian path in Königsberg? It’s simple: remove any one bridge. Interestingly, history created an Eulerian path of its own. During World War II, the Soviet Air Force destroyed two of the city’s bridges, making an Eulerian path possible, albeit unintentionally. These bombings effectively erased Königsberg from the map, and it was later rebuilt as the Russian city of Kaliningrad.
While Königsberg and its seven bridges may no longer exist, they are immortalized by the seemingly trivial puzzle that led to the birth of a new field of mathematics.
Draw a simplified graph of the Königsberg bridge problem. Represent each landmass as a node and each bridge as an edge. Count the degree of each node and discuss with your classmates why an Eulerian path is not possible with the original seven bridges.
Using the graph you drew, simulate the removal of one bridge. Identify if an Eulerian path is now possible. Discuss how the removal of a bridge changes the degree of nodes and the possibility of creating an Eulerian path.
Research a real-world application of graph theory, such as network design or urban planning. Present your findings to the class, highlighting how Euler’s insights into the Königsberg bridge problem apply to modern problems.
Design a new bridge problem using a different map layout. Exchange your problem with a classmate and attempt to solve each other’s puzzles. Determine if an Eulerian path or circuit is possible and explain your reasoning.
Explore the concept of Eulerian paths and circuits further by finding examples in puzzles or games, such as the game of “Snake” or maze puzzles. Share how these examples relate to Euler’s findings and graph theory.
Here’s a sanitized version of the transcript:
—
You’d have a hard time finding Königsberg on modern maps, but one particular quirk in its geography has made it one of the most famous cities in mathematics. The medieval German city lay on both sides of the Pregel River, with two large islands at the center. These islands were connected to each other and to the riverbanks by seven bridges.
Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town, became obsessed with these islands and bridges. He kept returning to a single question: Which route would allow someone to cross all seven bridges without crossing any of them more than once?
Think about it for a moment. Give up? It’s not possible. But attempting to explain why led the famous mathematician Leonhard Euler to invent a new field of mathematics. Ehler wrote to Euler for help with the problem. Initially, Euler dismissed the question as unrelated to math. However, the more he wrestled with it, the more it seemed there might be something significant after all.
The answer he developed was related to a type of geometry that did not quite exist yet, which he called the Geometry of Position, now known as Graph Theory. Euler’s first insight was that the route taken between entering an island or a riverbank and leaving it didn’t actually matter. Thus, the map could be simplified, with each of the four landmasses represented as a single point, known as a node, with lines, or edges, between them to represent the bridges.
This simplified graph allows us to easily count the degrees of each node, which is the number of bridges each landmass touches. Why do the degrees matter? According to the rules of the challenge, once travelers arrive at a landmass by one bridge, they must leave via a different bridge. In other words, the bridges leading to and from each node on any route must occur in distinct pairs, meaning that the number of bridges touching each landmass visited must be even. The only possible exceptions would be the locations of the beginning and end of the walk.
Looking at the graph, it becomes apparent that all four nodes have an odd degree. Therefore, no matter which path is chosen, at some point, a bridge will have to be crossed twice. Euler used this proof to formulate a general theory that applies to all graphs with two or more nodes. An Eulerian path that visits each edge only once is only possible in one of two scenarios. The first is when there are exactly two nodes of odd degree, meaning all the rest are even. In this case, the starting point is one of the odd nodes, and the endpoint is the other. The second scenario is when all the nodes are of even degree. Then, the Eulerian path will start and stop in the same location, which also makes it something called an Eulerian circuit.
So how might you create an Eulerian path in Königsberg? It’s simple: just remove any one bridge. Interestingly, history created a Eulerian path of its own. During World War II, the Soviet Air Force destroyed two of the city’s bridges, making a Eulerian path easily possible. Though, to be fair, that probably wasn’t their intention. These bombings effectively wiped Königsberg off the map, and it was later rebuilt as the Russian city of Kaliningrad.
So while Königsberg and its seven bridges may not be around anymore, they will be remembered throughout history by the seemingly trivial riddle that led to the emergence of a whole new field of mathematics.
—
This version maintains the content while ensuring clarity and coherence.
Mathematics – The abstract science of number, quantity, and space, either as abstract concepts (pure mathematics), or as applied to other disciplines such as physics and engineering (applied mathematics). – Mathematics plays a crucial role in the development of technology and the understanding of the universe.
Bridges – Structures built to span physical obstacles without closing the way underneath, often used as a metaphor in graph theory to describe connections between different parts of a graph. – In graph theory, bridges are edges that, when removed, increase the number of connected components of the graph.
Euler – Referring to Leonhard Euler, an influential Swiss mathematician and physicist who made significant contributions to various areas of mathematics, including graph theory. – Euler’s formula, V – E + F = 2, is a fundamental equation in topology, relating the number of vertices, edges, and faces of a polyhedron.
Graph – A collection of nodes (or vertices) and edges (or arcs) that connect pairs of nodes, used to model pairwise relations between objects. – The graph representing the social network showed how each person was connected to others through friendships.
Theory – A system of ideas intended to explain something, especially one based on general principles independent of the thing to be explained, often used in mathematics to describe a set of principles on which the practice of an activity is based. – The theory of relativity revolutionized our understanding of space, time, and gravity.
Nodes – Points in a graph where edges meet, representing entities or positions in a network. – In a computer network graph, nodes represent computers or devices, and edges represent the connections between them.
Degrees – The number of edges incident to a vertex in a graph, or a unit of measurement for angles in geometry. – In the graph, the vertex with the highest degree was the most connected node, indicating its importance in the network.
Path – A sequence of edges and vertices wherein a vertex is reachable from another vertex, often used to describe a route through a graph. – Finding the shortest path between two nodes in a graph is a common problem in computer science and logistics.
Geometry – The branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. – Euclidean geometry is based on the postulates of the ancient Greek mathematician Euclid, focusing on flat surfaces and shapes.
History – The study of past events, particularly in human affairs, often used to understand the development of mathematical concepts over time. – The history of mathematics reveals how different cultures contributed to the development of algebra and calculus.
