Bridges of Königsberg

Problem
Through the city of Königsberg flowed the Pregel River. In this river were two large islands, which were part of the city.

Joining the mainland either side of the river and those two islands there stood seven bridges.


 * [[File:Konigsberg_bridges.png]]

It was a popular exercise among the citizens to take a pleasure stroll across the bridges.

The question naturally arose: was it possible to cross each bridge once and once only during the course of a single walk?

Solution
The shape of the land and the positions of the bridges are irrelevant.

What is important is the relationships between the land and the bridges.

We can represent each of the landmasses as the vertices and the bridges as the edges of a multigraph:


 * KonigsbergBridgesGraph.png

The question now evolves into: does this graph allow the construction of an Eulerian path?

It can be seen that this is a graph in which there are four vertices, each of which has odd degree.

From Condition for Graph to be Traversable, it follows that this graph is not traversable, as it has more than two odd vertices.

The answer to the problem is, therefore: No.