The Seven Bridges of Königsberg.  The mathematical problem that changed the world of science

Königsberg is currently a Russian city founded at the mouth of the Peregula River. In the past, Prussia ruled this region, and the center itself was home to many famous mathematicians and physicists. Suffice it to mention Gustav Kirchhoff (creator of Kirchhoff’s Law), Carl Newman (creator of the concept of logarithmic potential), and Christian Goldbach (creator of Goldbach’s hypothesis that has not been solved yet).

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The secret of the seven bridges

A rather specific problem arose in this very science-friendly place. As we noted, the Prejula River flows through Krolevik, and due to its specific branching, The city was divided into four districts. To reach each of them, seven bridges were built: Kowalski Bridge, Connecting Bridge, Green Bridge, Merchant Bridge, Wooden Bridge, High Bridge, and Honey Bridge.

Photo: Merian-Erben/Wikicommons

The Pregula River divides Königsberg into four parts

While walking around Königsberg The residents devised a specific game whose goal was to find a way to cover the entire route, with the condition that each of the seven bridges could only be crossed once.. Many thought it was completely impossible, but no one had enough evidence to prove it. The issue became of interest to the mayor of Gdańsk, Gottlieb Eller, who wrote a letter to the Swiss mathematician and physicist on the subject. Leonarda Olera, participated in research at Saint Petersburg State University in Tsarist Russia. Euler thought the problem was trivial, but after taking a closer look at it, he became interested in it. This is what he wrote about to the Italian mathematician and engineer Giovanni Marinoni:

Euler believes that Gottfried Wilhelm Leibniz’s research on the so-called Position geometry. As it turned out, they became the basis for what we now call it Graph theory.

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A puzzle that cannot be solved

On August 26, 1735, Euler presented his paper on the question of bridges in Königsberg. The work entitled Solutio issuesatis ad Geometriam situs pertinentis, published six years later, presents the issue of this particular problem and a general solution for any number of bridges and places. The Swiss scientist, who studied Königsberg’s example, realized that it was accurate Town planning and the choice of route to follow have no influence on the possible solution of the problem. The only thing that matters is How specific elements relate to each other.

Euler transformed the map of the city into a simplified graph on which he placed dots to represent the four land masses, as well as the lines connecting them, namely seven bridges. And from this he noticed that If each connection (bridge) is to be crossed only once, each node (ground) must be accompanied by a pairwise connection list. One is used to enter the area and the other to exit it. A given node needs two connections if you visit it once, four if you visit it twice, and so on. Single nodes Which can have an odd number of links are those where the walk begins or ends.

A diagram showing the problem of bridges in KönigsbergPhoto: Mark Foski and Boyabazuka / WikiCommons

A diagram showing the problem of bridges in Königsberg

Knowing these relationships and looking at the map of Königsberg’s bridges, this becomes clear The puzzle cannot be solved. Every part of the city is connected Odd number of connections (Five for inside and three for others). Euler noted that there can be at most two areas with an odd number of connections to be able to complete the path according to the requirements of the puzzle.

Although Euler’s original proof seems simple, what is more important is how it is presented. Euler used an abstract view of the problem, using lines and letters to represent a complex situation. He assigned uppercase letters to lands and lowercase letters to bridges. In this way, he reduces a given puzzle to a set of vertices and edges. Today, a path in a graph that passes through each edge of the graph only once is called a path Euler Road.

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Graph theory was created thanks to Königsberg bridges

It is worth noting that graph theory itself currently has many applications in mathematics, electrical engineering, programming, networking, and even in fields such as sociology, marketing and communications. Furthermore, research suggests that it may provide insight into disturbances in information exchange between neurons. Which gives hope in combating progressive memory loss in Alzheimer’s disease.

Honey BridgeImage: public domain

Honey Bridge

We should also inform you that currently you will not be able to know whether Euler’s calculations are correct or not, because Most of the previously mentioned bridges in Königsberg no longer exist. Some were destroyed by bombing during World War II, others were demolished. Only the Drewniany and Miodowy bridges remain to this day.

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