In the early eighteenth century, there were seven bridges in the town of Königsberg (or Kaliningrad). They crossed the different branches of the River Pregel (or Pregolya), as shown in diagrammatic form in the figure. The question was asked whether it was possible, from some starting point, to cross each bridge exactly once and return to the starting point. This prompted Euler to consider the problem in more generality and to publish what can be thought of as the first research paper in graph theory. The original question asked, essentially, whether the graph shown is an Eulerian graph. It can be shown that a connected graph is Eulerian if and only if every vertex has even degree, and so the answer is that it is not.
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.