and the mathematician who figured this out was leonhard euler, who actually lived in konigsberg. he realized that the problem wasn't one about geography. it had nothing to do with the lengths of the bridges or their distances from one another, but it had everything to do with connectivity, which bridges are connected to which islands or riverbanks. he got to the essence of the problem by simplifying it, turning the bridges into lines, squashing the landmasses into points, which we'll call vertices. and in the process, he formed a graph, which became the topological backbone of the problem. euler had turned his trip around town into navigating a graph of four vertices and seven edges, thereby transforming the problem into one of graph theory, and thereby creating the field of graph theory and topology. in his solution, euler realized that, other than the vertices where you start and end your path, every time you take an edge into a vertex, you have to leave by another edge in order to avoid crossing the previous one again. so that means, if you're never going to double back, the n