About The Bridges of Koenigsberg The Bridges of Koenigsberg has been one of the more popular activities at the Family Math Nights hosted in the Ames, Iowa elementary schools. Walking the seven "bridges" is a large motor activity and has much appeal for active elementary students. The puzzle is an old one--the history is on the display. The mathematics Euler's solution of this puzzle marks the beginning of a new field of mathematics known as graph theory. It is used today for diverse applications including computer network design and analysis and planning shipping routes. The islands and bridges of the Koenigsberg bridge problem can be represented by a graph (the diagram shown on the module). The vertices (dots) represent the locations connected by bridges: that is, the banks and the islands. The edges (lines) represent the bridges. Euler proved that if more than 2 vertices have an odd number of edges meeting there, then it is impossible to trace each edge of the diagram exactly once without lifting the pencil. There are five bridges leading to Kneiphof Island, three to the other island, and three to each of the two banks. It's impossible to walk each bridge exactly once. You can extend this question: can you trace each edge of a graph exactly once and wind up back where you started? The answer to this is it will work as long as there is an even number of edges meeting at every vertex. (See the airline travel puzzle on the display board.) The materials You can make the bridges out of any large, readily available material. We used strips of cardboard from some discarded packaging and did a little artwork to make them look wooden. A couple of throw rugs represented the islands and some long runners borrowed from the school represented the banks. (Photo). Some related puzzles are provided in Printable Material to go with the main activity but it is easy to find alternatives: these graph puzzles are very popular in children's puzzle books. A couple of good sources are Amazing Math Puzzles by Jeff Sinclair and The Little Giant Book of Math Puzzles by Derrick Neiderman. It would be a good idea to have a couple of pencils and some scratch paper available for solving the puzzles. A hand-sketched map of the part of Eastern Europe containing Koenigsberg is provided in Printable Material. For an official map of the Baltic region look at this website: http://www-groups.dcs.st-andrews.ac.uk/~history/BirthplaceMaps/Places/Germany.html A map of Koenigsberg as it was in the 18th century: http://www-groups.dcs.st-andrews.ac.uk/~history/Diagrams/Konigsberg_colour.jpeg Kaliningrad (formerly Koenigsberg) as it is now: http://www-groups.dcs.st-andrews.ac.uk/~history/Diagrams/Kaliningrad_now.jpeg Staffing the display The volunteers who worked with children on this display seemed to be very busy throughout the evening due to the amount of interest the activity generated. It is useful to have someone available to oversee this activity, and it's a good idea for the staffer to have done the puzzles before kids arrive. Not too much guidance is needed--usually kids will take a trial-and-error approach to it and try to work it out on their own. Traffic control is needed, though. Some interesting websites: Graph theory activities: http://www.math.ucalgary.ca/~laf/colorful/colorful.html History of topology, beginning with Euler's solution to Koenigsburg Bridges: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Topology_in_mathematics.html Background on the Koenigsberg Bridge problem, followed by related interactive puzzles: http://www.cut-the-knot.com/do_you_know/graphs.html Written by Janet A. Dixon Jan 5, 2002 Revised 1/10/06