The Königsberg bridges cannot all be traversed in a single trip without doubling back. This problem was solved by Euler, and represented the beginning of Graph Theory.
See also Eulerian Circuit, Graph Theory
References
Bogomolny, A. ``Graphs.''
http://www.cut-the-knot.com/do_you_know/graphs.html.
Chartrand, G. ``The Königsberg Bridge Problem: An Introduction to Eulerian Graphs.'' §3.1 in
Introductory Graph Theory. New York: Dover, pp. 51-66, 1985.
Kraitchik, M. §8.4.1 in Mathematical Recreations. New York: W. W. Norton, pp. 209-211, 1942.
Newman, J. ``Leonhard Euler and the Königsberg Bridges.'' Sci. Amer. 189, 66-70, 1953.
Pappas, T. ``Königsberg Bridge Problem & Topology.'' The Joy of Mathematics.
San Carlos, CA: Wide World Publ./Tetra, pp. 124-125, 1989.