In the early 18th century, there were seven bridges in the town of Königsberg (now Kaliningrad). They crossed the different branches of a river as shown in the figure. The question was asked whether it was possible, on a single walk, to cross each bridge exactly once. This prompted Euler to consider the problem more generally and publish arguably the first research paper in graph theory. The original question asked, essentially, whether the graph shown is traversable. It can be shown that a connected graph is traversable if and only if there are 0 or 2 vertices with odd degree. Compare Eulerian trail.

A map of the bridges

The bridges as a graph

http://nrich.maths.org/public/viewer.php?obj_id=2327 An interactive page in which you can construct your own problem like the bridges of Königsberg.