Bridges of Knigsberg & Eulerian circuits

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Bridges of Königsberg & Eulerian circuits

The 'Königsberg bridge' problem is based on a city that had seven bridges, which connected two islands. The problem was to find whether there is any way to cross over all seven bridges once without turning back onto a bridge that has already been crossed.1 Figure 1 shows what the bridges and river looked like when Eular was around.

Fig. 1

In, 1736 Euler the problem using what people now believe as the start of graph theory. He proved that there were no solutions at all to this problem hence you cannot cross all the bridges exactly once. The way Euler solved it was by thinking of the river and bridges in terms of graph theory. This was achieved by removing anything unnecessary to the problem and so therefore ended up drawing a picture primarily consisting of points that represented the islands and separate lines showing the bridges that connected the islands.2 Figure 2 shows the picture of what Eular might have drawn up.

Fig. 2

As you can see the problem now looks much simpler and one could attempt to solve by just using a pencil and going