Math 7
Prof. Winkler
Ethan Klein
4/4/13
The Seven Bridges of Konigsberg
When asked to write about my favorite mathematical puzzle, I immediately thought of
The Seven Bridges of Konigsberg. This puzzle stands out in my mind as a stellar example of
what a good puzzle or problem should be. It is simple to understand, surprisingly difficult to
solve, and the process of solving it leads one to conclusions of far greater import than the puzzle
would ever initially appear to reveal. As an added bonus, the puzzle is also of great historical
importance, essentially founding the fields of topology and graph theory.
The original statement of the puzzle, as the great mathematician Leonhard Euler solved it,
is something along the lines of the following:
The King of Prussia is planning a grand parade through the streets of Konigsberg for the
first annual celebration of his coronation. Being both lazy and rich, the king intends to
watch the parade from his private river yacht. Since he wants to see the parade from
every possible viewpoint, he insists that the parade cross over all seven bridges in the city.
However, as you may have guessed, the king isn’t very good at managing the treasury,
and only has enough money for the boatman to watch the parade go over each bridge
once. Additionally, for logistical reasons, the parade must begin and end in the same
place. Using the map below, can you find a route that satisfies the king’s demands?
Konigsberg c. 1735, with the bridges and river highlighted
While this is the classic statement of the problem, it is sadly out of date. In World War II,
Konigsberg became a heavily fortified Nazi holdout during the long final Soviet offensive on the
Eastern front. Neither the Nazis nor the Soviets are especially well-known for their great respect
of mathematical history, and, after the battle, only four bridges remained in Konigsberg (later
renamed Kaliningrad). (Taylor) With the original motivation for the puzzle literally blown to
pieces, it seems that an alternate statement of the problem is in order. Here is a puzzle which is,
perhaps surprisingly, equivalent, and serves to illuminate the broader import of solving the
Konigsberg Bridge Problem.
A very bored researcher working with jumping spiders is killing time by making them
jump across posts between the glass windows of an otherwise empty ant farm. When a
spider jumps from one post to another, it trails some web attached to the first post behind
it as a safety line. When they land, the spiders attach their web lines to the landing post,
leaving a line hanging between the two. The spiders will not jump between the same two
posts twice, as they would become tangled the webbing left hanging. The researcher in
question is trying to make the spiders form particular shapes with spent webbing. Without
attempting to coax a spider to crawl back along web it has already used, is it possible for
a single spider to make the figure below, starting and ending at the same post?
Bear with me for a second if you don’t immediately see the equivalency. By switching
from paths between islands to essentially paths between a set of points, this statement of our
puzzle clearly illustrates a crucial idea to the solution which the original presentation almost
actively attempts to obscure: as far as this problem is concerned, it is irrelevant how one moves
on the connected spaces, be they posts or islands. The only important thing is the order in which
one moves over those connections. This allows us to abstract the map of Konigsberg, or the
spider’s web, to a general graph comprised of “nodes” and “edges” connecting them. We find to
our delight that they yield the same graph, as shown below.
Hopefully, the equivalence between these two statements of the same problem has
become clear. All we have to do to solve both problems is find a way to trace the above graph
without lifting the pencil or retracing our path, or to show that this is impossible. From this
generalized statement of the puzzle, we can see the wonderfully elegant solution.
The first step in solving this puzzle is simply to note that when the line passes through a
node, it draws two edges, but only draws one when leaving the starting node or stopping at the
last node. Thus, a figure can only be drawn in a single stroke by starting or ending at any node
with an odd number of edges. Since we cannot pick up our pencil, this means that only figures
which can be drawn in a single stroke have no more than two nodes with an odd number of edges.
Additionally, if we wish to stop at the same point we started from, all nodes in the graph must
have an even number of edges. It’s that blissfully simple.
It’s simply astounding what Euler accomplished here by solving one problem. He crossed
The Seven Bridges of Konigsberg to discover on the other side a geometry independent of any
notion of metric or measure. This laid the foundation for future work in what he called
geometriam sitvs (geometry of position), now known as topology, as well as graph theory,
crucial in myriad fields such as computer science and sociology. That one seemingly
unremarkable mathematical quandary brought about such a wave of discovery is more than a
little awe-inspiring, and that’s why The Seven Bridges of Konigsberg will always have a very
special place in my heart.
Works Cited
Taylor, Peter. "What Ever Happened To Those Bridges?" Australian Mathematics Trust. N.p., 2007. Web.
4 Apr. 2013. <http://www.amt.edu.au/koenigs.html>.