3 Lessons from Konigsberg Bridge Problem

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The Konigsberg Bridge Problem originated in the erstwhile city of Konigsberg, a Baltic Port, which was located in the Kingdom of Prussia and Germany until 1946.

Konigsberg was founded in 1255 by the Teutonic Knights after their second crusade against the Prussians and was named in honor of King Ottokar II of Bohemia. Konigsberg then became the capital of the Duchy of Prussia (1525-1701) and East Prussia (until 1945). Now it is known as Kaliningrad and part of Russia.

The Pregel river flowed through Konigsberg around two islands, Kneiphof and another island and divided the city into four distinct regions. During the Middle Ages, Konigsberg was an important port town and maritime trading center. It was a flourishing port with a thriving economy which allowed the construction of seven bridges across the river and two islands.

Konigsberg Bridge Problem Map
Konigsberg Map | Source : http://www-history.mcs.st-andrews.ac.uk/Extras/Konigsberg.html

Konigsberg Bridge Problem

The locals who were quite rich and could afford a lot of leisure time devised a problem which later came to be known as the Konigsberg Bridge Problem.

It asked if anyone could walk around the entire town in such a way that each bridge was crossed only once.

There are actually two aspects to the Konigsberg Bridge Problem. It tries to see if anyone cross all the seven bridges in a single trip,
1. without doubling back,
2. and ending in the same place where he began.

Many people tried to solve this seemingly easy problem, and they instinctively knew it was not possible. But they also could not prove that it wasn’t possible.

In 1736, Leonhard Euler, a Swiss mathematician proved that it was not possible to do such a walk. He in fact went on to devise a whole new branch of mathematics, now known as Graph Theory and heavily used in modern Computer Science.

So now that we know what the Konigsberg Bridge Problem is, let us see what we can learn from it.

The Three Lessons

Hidden behind every trivial problem lies a complex and challenging one.

As seen above, the Konigsberg Bridge problem started out as a trivial and simplistic problem. What could be more mundane than strolling on bridges in a city and coming back to the starting point?

But for some reason it piqued Euler’s interest and he went on to say the following.
This question is so banal, but seemed to me worthy of attention in that [neither] geometry, nor algebra, nor even the art of counting was sufficient to solve it.

In life we come across seemingly simplistic and banal problems, but for some reason we are not able to find simple solutions.

We must therefore appreciate the fact that hidden behind the easy is the very difficult and modify our thinking accordingly.

Sometimes we must step back and view a problem abstractly.

Euler viewed the Konigsberg Bridge problem abstractly. He converted bridges and landmasses into notations now known as edges and vertices in graph theory.

For those interested, mathematically the Konigsberg Bridge Problem can be stated as below. “Does the multigraph on four nodes and seven edges have an Eulerian cycle?

Sometime when we are in the midst of something, we miss the wood for the trees. We must step back and take a birds-eye view to ensure that we do not get bogged down by the minute details and in that process miss the big picture.

An apparently simple problem can provide us a template to solve complicated problems.

The Konigsberg Bridge problem led Euler to invent a new system of mathematics known as Graph theory which now find application in modeling relations and processes in physical, social, biological and computer science.

Same happens in life also.

That’s it then. Do share your comments and let me know your thoughts. Thanks in advance for your comments and likes.

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Reference

NRICH Project
Wolfram
Wikipedia
MAA

Konigsberg Map : http://www-history.mcs.st-andrews.ac.uk/Extras/Konigsberg.html

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