Solving The Unsolvable – The Seven Bridges of Konigsberg: Euler’s Path To Genius

Math was never my strong suit. Basic Algebra was the limit of my arithmetical competence. Everything beyond that was a struggle. In geometry I struggled to a grade of C, Algebra II a grade of D and I dropped Trigonometry after one week. I knew getting through any university level math courses would be a struggle. Imagine my surprise then, after I got to college and found a course called Infinite Math. It was not nearly as daunting as its name. The course consisted of Maths that could be applied to the real world. My favorite of these was something called Euler Circuits, which meant finding the most efficient route for a journey. There was also the less rigorous Euler Path. Trying to figure out the most efficient Euler Circuit or Path became one of my favorite mathematical exercises.  As for the name Euler, it never meant much to me until I recognized it again after many years while reading about Konigsberg, the old capital of Prussia and today the city of Kaliningrad, in the Russian oblast of the same name.

The Seven Bridges of Konigsberg

The Seven Bridges of Konigsberg

A Problem Without A Solution – Explaining The Impossible
The Old Town of Konigsberg stood on both sides of the Pregel River. Uniquely, as the river flowed through the city it wove its way around two islands. The more famous of the two was the Kneipfhof which had five bridges going across arms of the river. Another two bridges crossed branches of the river from another island, Lomse. These seven bridges were the genesis of a puzzle that many in the town tried to solve. As one resident of Konigsberg related in a letter to Swiss mathematician Leonhard Euler, couples in the town liked to try and figure out a route to cross every bridge once without ever having to re-cross any of the same bridges again. An even tougher problem would be to do this while ending up back in the same place they began. In 1736 Euler set himself the task of proving that a solution to this problem was impossible. He did this by focusing only on the land masses and bridges. He made each land mass a “point” or in modern parlance a “node”. Each connecting bridge was an “arc”. This abstraction could then be drawn as a graph. Euler’s proof was published in 1741, six years after he first began to study the bridges problem.

The essence of the problem was how to draw this upon a sheet of paper without retracing any line or lifting a pencil off the paper. This laid the basis for the first ever theorem of graph theory. Euler’s name was given to among other things, Euler Paths which is a continuous route that passes every edge once and only once. His name was also given to Euler Circuits, a path beginning and ending at the same starting point without retracing any part of the route. Many people in Konigsberg understood from experience that there was no route that could be followed to cross all Seven Bridges of Konigsberg once and only once without retracing some part of the route. Euler’s innovation was that he could explain the impossibility of a solution and used it to develop the basis for graphs, networks and topology. Euler’s mathematical genius extended to the counter-intuitive. With the Seven Bridges of Konigsberg he proved the rationale, reasoning and intellectual uses of a problem that could never be solved.

The Wooden Bridge - in 1930's Konigsberg

The Wooden Bridge – in 1930’s Konigsberg (Credit: Eigenes Werk)

Bridging A Divide – From Konigsberg To Kaliningrad
Crossing the Seven Bridges of Konigsberg as Euler knew them is impossible today, but not because of any mathematical problems. The difficulty arises from the fact that, like almost all of old Konigsberg, most of the bridges no longer exist in their original form. Two of the bridges – Blacksmith’s Bridge and Giblet’s Bridge – were destroyed in the British bombing of the city. Both of those bridges led to and from Kant Island. The bombing which took place on two nights in late August of 1944, also leveled much of the castle and cathedral, though the latter has been rebuilt. Two other bridges – the Shopkeeper and Green Bridge – disappeared after the war to make way for Leninsky Prospekt in what had suddenly become Kaliningrad, a closed Soviet city. Thus, the Seven Bridges of Konigsberg were now three bridges in Kaliningrad. The most popular of the three that still exists today is also the only one that goes to Kneiphof, the aptly named Honey Bridge. Like many other famous bridges in European metropolises it sports hundreds of padlocks which are symbols of those romantic couples hoping these symbols will secure their love forever.

The Honey Bridge leads between the reconstructed Cathedral and the Fishing Village, two of the most famous spots in the Old Town which only adds to the foot traffic. Another bridge which is original, the Wooden Bridge, was lucky enough to escape destruction by either bombs or Bolsheviks. For historical harmony, it would be nice if all the bridges were rebuilt, only one holds that honor and it was rebuilt before the war by Germans, not afterwards by the Soviets. It is known simply as the High Bridge. The upshot of all this bridge building, crossing and destruction is that only two of the seven bridges that existed during Euler’s time can still be found in their original form today. It is easy enough to cross those two bridges without having to retrace one’s footsteps. Yet there were and still are many more bridges in Konigsberg to cross, perhaps not as famous, but just as important in their own way.

Leonhard Euler - mathematical genius

Leonhard Euler – mathematical genius (Credit: Jakob Emanuel Handmann)

Fits Of Mathematical Imagination – The Seven Bridges Of Kaliningrad
One can only speculate as to all the different problems and solutions Euler could have concocted by adding or subtracting these bridges in his theoretical fits of mathematical imagination. Today Euler would have the option of adding the newly built or refurbished Flyover and Jubilee Bridges to his equation. This has brought the total of bridges in the city back up to seven. Many things have changed in Kaliningrad and Konigsberg is no more, but the Seven Bridges problem still exists, albeit in extremely modified form.

 

1 thought on “Solving The Unsolvable – The Seven Bridges of Konigsberg: Euler’s Path To Genius

  1. I was first introduced to the seven bridges problem half a century ago by our maths teacher, and have never forgotten it. Good to read an illustrated post about it and the actual city, too.

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