Seven Bridges of Königsberg
In modern language, Euler shows that whether a walk through a graph crossing each edge once is possible or not depends on the degrees of the nodes. The degree of a node is the number of edges touching it. Euler shows that a necessary condition for the walk is that the graph be connected and have exactly zero or two nodes of odd degree. This result stated by Euler was later proved by Carl Hierholzer. Such a walk is now called an Eulerian path or Euler walk. If there are nodes of odd degree, then any Eulerian path will start at one of them and end at the other. Since the graph representing the historical Königsberg has four nodes of odd degree, it cannot have an Eulerian path.