Is it possible for a visitor to Kaliningrad to tour all seven bridges of this city so he/she returns at the beginning, crossing each bridge just once and only once? Although even things that are possible appear completely impossible, not to mention such a common thing as taking a walk over seven bridges, the answer to the question is that it is not possible.

This famous mathematical puzzle is known as the Königsberg bridge problem, which was Kaliningrad’s name until it was renamed in 1946, having become a part of the then-Soviet Union. Königsberg bridge problem was solved in 1735 by the Swiss mathematician Leonhard Euler (1707–1783). His solution is pretty renowned in mathematics since it practically led to the development of graph theory and topology, nowadays rather advanced mathematical disciplines, which paved the way for applications in physics and a range of other scientific fields.

The Pregel (Pregolya) River runs through Kaliningrad, i.e. Königsberg, which circles two big islands in the very centre. To interconnect islands, as well as the left and right banks, seven bridges were built across the river many centuries ago, as shown in this 16^{th}-century graphic.

When Euler posed the question, two and a half a century ago, on how to walk the bridges crossing them once and only once, the first step toward solving this puzzle was not to start touring the bridges, but to draw a graph on which each landmass would be presented by a node and each bridge by a line.

Helped by this graph, which was a harbinger of new math disciplines, he determined whether an even or an odd number of lines entered a node, and thus he concluded that the so-called Euler’s spiral cannot be drawn through the graph. He proved that the bridges of Königsberg cannot be toured by crossing them only once.

Königsberg i.e. Kaliningrad is not only famous for this remarkable Euler’s puzzle but also because one of the greatest German philosophers, who both lived and came up with his philosophical theory, and who never left Königsberg, Immanuel Kant (1724–1804) was born here.

The city is currently part of a smaller area of Russia, which is, due to historical circumstances, completely separated from the homeland and is located among Germany, Lithuania and the Baltic Sea. It used to belong to Prussia and following the Potsdam Conference, on Stalin’s request, it was given, the city and the adjacent area, to the Union of Soviet Socialist Republics, after which the Soviets expelled the German population.

Having been separated from mainland Russia since 1990, this area is underdeveloped, even partially deserted, but the Russian Federation changed its tune around 10 years ago and made Kaliningrad a special duty-free zone (dubbed European Hong Kong), which in recent years has brought about an extraordinary economic boost.

Kaliningrad was heavily bombed during the Second World War, so the famous seven bridges from Euler’s puzzle have not existed for a very long time – two of them were destroyed in bombings.

S.B.

Illustration: Historic Cities Research Project