27/07/2022

In a column in Folha, Viana explains the mystery of the seven bridges.

Cidade de Königsberg | Foto: Wikimedia Commons

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

I like to tell this story to explain why mathematics is much more than "the science of numbers," because it's a tale of beautiful mathematical ideas that have little or nothing to do with numbers. Ironically, the hero, the great Leonhard Euler (1707–1783), had a different opinion. But we have three centuries of head start over him.

The story begins in 1255, when the city of Königsberg was founded by the Teutonic Knights at a fork in the Pregel River. The name ("King's Mountain" in German) was a tribute to King Ottokar II of Bohemia. A transit point for the rich Baltic Sea trade, Königsberg soon became one of the most prosperous Germanic cities and, later, the capital of the Duchy of Prussia.

At the beginning of the 18th century, the settlement extended across four regions—the two banks of the Pregel River, the island of Kneiphof, and the landmass (Lomse) between the river branches upstream from the fork—connected by seven bridges: two bridges to Kneiphof and one to Lomse on each bank, plus one bridge connecting Kneiphof to Lomse.

Euler was not impressed. In his reply to his colleague in March 1736, he wrote: "You must understand, noble sir, that this type of solution has little to do with mathematics. I do not understand why you expect a mathematician to provide it, and not someone else, since the solution is based solely on reasoning, and the discovery does not depend on any mathematical principle."

But by that time, Euler had already submitted the solution for publication in the annals of the St. Petersburg Academy of Sciences (it would only be published in 1741). Far beyond solving the Königsberg problem, this work founded a new mathematical discipline, graph theory.

To read the full text, visit the newspaper's website.