Königsberg Bridge Problem
The problem is simply to find a path around a collection of bridges that crosses each
bridge exactly once. The map below shows the layout of bridges in the town of
Königsberg, with seven bridges and two islands.
Map of Königsberg
The seven bridges problem is well-known in both Maths and Computer Science, and so is
particularly appropriate for the garden since the bridges problem links the two
departments that sponsored the garden! It is usually associated with the 18th century
mathematician Leonhard Euler -- a path that solves the problem is called an Eulerian
path. Eulerian paths have interesting properties, both mathematically and
computationally. The layout of bridges is from the town of Königsberg in northern
Germany (now part of Russia). The islands are in the middle of a river. In our garden the
river is represented by the garden. Since it is a river, you aren't allowed to go around the
far ends of the diamond!
Euler is supposed to have proved that there is no solution when he was working in
St. Petersberg. There could only be a solution if every island had an even number of
bridges touching it, since one must leave an island the same number of times one arrives
at it. Alternatively, exactly two islands can have an odd number of bridges, and these
must be the start and finish point of the tour.
Related sites
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http://forum.swarthmore.edu/~isaac/problems/bridges1.html
http://history.math.csusb.edu/HistTopics/Mathematical_games.html (includes a
map of the original town of Königsberg).
http://www.cate.org/euler.htm
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Queen's College, Cambridge have a Mathematical Bridge; the mathematics in this
case apparently relates to the design of the bridge, and finding an Eulerian path is
not so difficult in this case!
Oh, and the Eagles have a song called Seven Bridges Road, which is completely
unrelated!
Back to the Bridges of Friendship Garden Homepage
Modified: 6-Apr-1999
http://bridges.canterbury.ac.nz/features/bridges.html
Here is an olde drawing of Königsberg:
http://www-groups.dcs.st-and.ac.uk/~history/Diagrams/Konigsberg_colour.jpeg
Leonard Euler’s biography is fascnating. You can look up information at http://wwwgroups.dcs.st-and.ac.uk/~history/Mathematicians/Euler.html
The Konigsberg Bridge Problem
In the first paper on Graph/Network Theory, written in 1736, Leonhard
Euler considered the following problem:
The residents of the Town of Konigsberg (now Kalinigrad) had seven
bridges over the Pregel River as schematically drawn below.
The question raised by Euler was: Would it be possible to start at any
place in the town, cross each bridge exactly one time, and return to
the starting place? Either construct such a tour of Koningsburg or
prove that it would be impossible to do so.
By constructing the graph below to represent the problem, Euler
concluded that it was impossible.
http://supernet.som.umass.edu/facts/kbridge.html
FROM DETAILED REALITY to the BARE BONES of a GRAPH:
First, Euler mathematically represented the problem by formalizing the system. In figure
1 we see a cartographer's rendition of Konigsberg drawn from a helicopter over the city
in 1739. In figure 2, we see how Euler visualized this same map. You may at first think
that Euler was actually completely blind before 1766 (the year in which Euler did in fact
go blind). However, figure 2 is a simplification of the problem called a graph. A graph is
a system of vertices connected by paths. An Euler path is a continuous path that traverses
the paths (starting and ending on a vertex), passing over each path exactly once. An Euler
circuit is an Euler path which begins and ends at the same vertex. Ideally, the residents of
Konigsberg could have taken an Euler circuit stroll around their city, crossing each
beautiful bridge once and no more, and ending up safely at home upon concluding their
promenade. More realistically, the natives simply hoped to find an Euler path (not
necessarily a circuit) on which they did not have to cross the beautiful-once, but not-sobeautiful-the-second-time-around bridges only a single time, and ultimately cross just one
or two bridges (for the second time) after following the completed path to get home
again. Lastly, they hoped, if all else was futile, to at least prove that such a path was
impossible, thus ending years of bitter debate and Sisyphean wandering.
Euler labored and exposed the truth. First he divided the vertices into odd and even based
on the parity of the number of paths directly connected to that vertex. He then noted
about any graph (such as the one in figure 2) the following four rules:
1. There will always be an even number of odd vertices.
2. If there are no odd vertices, then there is an Euler circuit starting at every vertex.
3. If there are two odd vertices, then there is no Euler circuit, but at least one Euler
path starting at one odd vertex and ending at the other.
4. If there are four or more vertices, then there are no Euler paths.
( Exercise: informally reason about the truth of these 4 rules. ) Upon carefully
scrutinizing the graph in figure 2, the forlorn citizens of Konigsberg gave up their
impossible search, Euler having thus shown that their hoped-for path could not exist by
rule (4).
http://members.aol.com/tylern7/math/euler-8.html
MATHEMATIZATION: A WALK IN THE PARK
by: Susan K. Eddins
Illinois Mathematics and Science Academy
There is a park, Jardin de la Ville Amélia, in Barcelona, Spain, where most mornings the
older men of the neighborhood meet to walk and visit. While walking in this park with a
Spanish colleague during a conference this past fall, I was reminded of Euler's famous
Königsberg Bridge Problem of 1736 which is often cited as the starting point for the
mathematical field known as graph theory.
The German town of Königsberg was built on both sides of a river. The town included
two islands connected to the shores and each other by a series of seven bridges. The
question posed by Euler was whether residents of Königsberg could take a stroll in the
evening during which they would cross each of the seven bridges exactly one time. To
answer this question, Euler mathematized it, that is, he took the essential elements of the
situation and represented them using mathematical objects. In this case he used segments
of lines and curves and their points of intersection.
Euler chose to represent each land mass by a point, called a vertex, and each bridge by a
segment, called an edge. When mathematized in this way, Euler's question is the same as
asking whether it is possible to trace over every segment of this network exactly once
without lifting your pencil.
Euler reasoned as follows: If the degree of a vertex is odd, that is, if it has an odd number
of edges coming into it, then you must either begin or end your tracing at this vertex. To
approach a vertex along one edge and leave along another requires a pair of edges.
Because all four vertices in the Königsberg network are of odd degree, traveling across
each bridge exactly one time is not possible.
The challenge presented to you is to use the drawing of the Jardin de la Ville Amélia to
mathematize the situation and to determine whether Dr. Perez-Pardo could walk along
every path in the park exactly one time during his morning stroll. If it is possible, where
must he begin and end? If it is not possible in one walk, how many different walks would
it take for him to be able to do so and where could he begin and end each? [You are
invited to mail your solutions along with your comments back to IMSA.]
Paths in the Jardin de la Ville Amélia
[Hint: You may need to think carefully about what you represent as vertices.]
Reference
Kenney, Margaret J. (editor), Discrete Mathematics Across the Curriculum, (the 1991
Yearbook of the National Council of Teachers of Mathematics), NCTM, Reston, VA,
1991.
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