Königsberger Brücken
About this app
Stilisiertes Königsberg 1736 mit Pregel, Kneiphof und sieben Brücken: Die Spazierfigur über die Brücken ziehen, bis die Sackgasse kommt. Der Euler-Blick (mit Vorhersage zum Abstimmen) lässt den Stadtplan zum Graphen schrumpfen, paart an jedem Ort die Brücken und lässt die Orte mit Rest leuchten. Brücken bauen und abreißen, Beweis-Modus in sechs Schritten, Haus vom Nikolaus und eigene Graphen.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
The bridge problem and Euler’s theorem
In Königsberg in 1736, seven bridges joined four parts of the city on the river Pregel: Old Town, Suburb, the island Kneiphof and Lomse. Is there a walk that uses every bridge exactly once? Leonhard Euler proved that there is none. He turned the city map into a graph: places become points, bridges become edges. A walk over every edge exactly once exists exactly when the edges are connected and at most two points have an odd number of edges. For a round trip there may be no such point. In Königsberg all four places have an odd number of bridges.
What the app shows
A stylised city map shows the Pregel, Kneiphof with its cathedral, Lomse and the seven bridges in their historical positions. A walker crosses the bridges, walked bridges change colour and a dead end becomes visible. The Euler view shrinks the land masses into points and turns the bridges into edges. At every place, arcs join the bridges into pairs of “in” and “out”; if one is left over, the place glows. A proof mode walks through Euler’s argument in six steps.
Try it yourself
Drag the walker across the bridges and try to use each exactly once. Before the Euler view, a prediction card asks whether it is possible, so a class can vote. In build mode you place an eighth bridge on the water or tear one down and see at once which places still have a leftover. Further presets are Kaliningrad today, the House of Santa Claus and your own graph. Under Practice, challenges are waiting, such as a round trip with as few new bridges as possible.
Frequently asked questions
- What is the Seven Bridges of Königsberg problem?
- The question whether you can take a walk in Königsberg that crosses each of the seven bridges over the river Pregel exactly once. Leonhard Euler proved in 1736 that this is impossible. His solution is regarded as the beginning of graph theory and topology.
- When does an Euler path exist?
- An Euler path uses every edge of a graph exactly once. It exists exactly when the edges are connected and at most two vertices have odd degree, that is an odd number of edges. The path then starts at one of these vertices and ends at the other. If all vertices have even degree, there is even a round trip.
- Why does an Euler path need even degrees?
- Whoever passes through a place on the way comes in over one edge and leaves over another. Every pass uses two edges, so the edges of a place can be paired. An edge without a partner can only be at the start or the end, which is why at most two places may have an odd number of edges.
- What can I try in the app?
- Drag the walker across the bridges, shrink the city map into a graph, build and tear down bridges, step through Euler’s proof, draw the House of Santa Claus in one stroke and solve challenges, for example placing an eighth bridge so that the walk becomes possible.
Subject: Mathe ohne Zahlen
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