A city, an island, seven bridges
The question at stake was concrete and local: could a person walk through the city of Königsberg, which straddled the Pregel River and included two islands connected to the mainland and to each other by seven bridges in total, crossing every single bridge exactly once during a single walk. Leonhard Euler presented his answer, a proof that no such walk exists, to the St Petersburg Academy on 26 August 1735, later publishing it in 1741. His method was to strip away everything about the problem that did not actually matter: the exact route taken within any given landmass was irrelevant, since only the sequence in which bridges were crossed determined success or failure, so he represented each landmass as a single point and each bridge as a line connecting two points, creating the first structure of what mathematicians now call a graph.
Throwing away the map
Working with this simplified representation, Euler proved a general rule: a walk crossing every connecting line exactly once can exist in such a structure only if exactly zero or exactly two of the points touch an odd number of connecting lines. In Königsberg’s specific case, all four landmasses touched an odd number of bridges, with one landmass connected by five bridges and the other three each connected by three, meaning the city’s bridge network violated the rule on every single point rather than merely one, ruling the desired walk out completely rather than merely making it difficult to find. The proof did not require Euler to attempt and fail at finding a route; it demonstrated, from the structure of the connections alone, that no route could possibly succeed.
A rule about odd numbers
What has held up, more than the specific bridge puzzle itself, is the value of Euler’s underlying abstraction. His insight that a network’s physical layout is irrelevant to certain of its most fundamental properties is now credited as the first theorem of graph theory and the first true proof in network theory, and it anticipated the later development of topology, the branch of mathematics concerned with properties that survive continuous deformation of a shape. Graph theory grew substantially in the century that followed: Gustav Kirchhoff applied related mathematics to electrical circuits in 1845, Arthur Cayley used graph structures to study chemical compounds in the 1870s, and the field eventually acquired its first dedicated textbook, by Dénes Kőnig, in 1936, with Frank Harary’s 1969 text later becoming its standard reference.
The first theorem of a new field
What has not survived, ironically, is the actual puzzle Euler solved. Of the seven original bridges, two were destroyed by bombing during the Second World War, and two more were later demolished to make way for a highway, leaving three of the original seven still standing, only two of which date from Euler’s own era, with the third rebuilt in 1935. This matters because the network Euler proved impossible to traverse no longer exists in its original form: with only five bridges remaining rather than seven, the graph representing modern Königsberg, now the city of Kaliningrad, has only two landmasses with an odd number of connecting bridges rather than four, meaning the very walk Euler proved impossible in 1735 has, purely as an accident of wartime destruction and later urban planning, become theoretically achievable today.
The bridges that no longer exist
The reach of Euler’s abstraction now extends into fields he could not have anticipated. Graph structures describe the link networks underlying websites and social media platforms, guide the physical layout of computer chips, and model molecules directly, with atoms represented as points and chemical bonds as connecting lines, a direct descendant of Cayley’s nineteenth-century work applying similar structures to chemistry. In biology, the same framework is used to map metabolic pathways, gene regulatory networks, and the connections between neurons in connectomics research, while in the social sciences, graph-based network analysis is used to measure an individual’s influence or prestige within a group by examining the structure of their connections rather than any property of the individuals themselves. All of this traces back to a decision to represent a city as nothing more than points and lines.
The puzzle solved by accident
Yes, and it is a genuinely satisfying case where a small, almost recreational puzzle turns out to have opened an entire mathematical field rather than remaining a curiosity. Working through the impossibility yourself, trying and failing to find the walk before understanding why it cannot exist, gives real weight to the abstraction Euler introduced, since the leap from a specific city’s bridges to a general rule about points and lines is not obvious in advance. The additional twist, that the physical bridges have since changed enough to make the original puzzle solvable after all, adds a rare kind of historical irony to a piece of mathematics that is otherwise valued for its permanence rather than its accidents.