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The Bridges of Konigsberg

Age 11 to 18
Challenge Level

We received three good solutions - two almost completely correct, fromYanqing from Lipson Community College and from Sarah from Colyton, and one completely correct from Andrei from Tudor Vianu College in Bucharest. Well done to you all.

Andrei's solution follows:

First, I observe that for the first figure it is impossible to cross each bridge once. This happens because you would remain in a place where all the bridges were already crossed.

In the second case, it is possible to cross the bridges only once and create a circuit.

I worked with different number of bridges and islands and I obtained solutions for each of the following:
- bridge traversing circuits
- bridge traversing paths
- Hamiltonian circuits
- Situations where none of the above is possible.

For 8 bridges I found:

- A diagram with a bridge traversing circuit.

a bridge traversing circuit
The circuit is the following: S-C-A-D-B-A-C-B-S

- A diagram with a bridge traversing path:

A bridge traversing path

The path is the following: S-D-E-A-B-S-C-E-B.

- A Hamiltonian circuit and a bridge traversing circuit:

A Hamiltonian circuit

- A diagram in which neither is possible. In the diagram below, neither a traversing path nor a traversing circuit is possible:

non traversable diagram

Editor's note:
If you were allowed to start at another island you could have a traversable path: starting at the top right hand island and finishing in the top left hand island, or vice versa.