- K:)nigsberg

# K:)nigsberg bridges solved

The final years of the last millennium saw the solutions of two out of the three most famous mathematical problems: the map-coloring problem and Fermat's last theorem. The time was ripe to attack the third: The problem of the Königsberg bridges. The problem is known as Eulerian trail and it was formulated by Euler as a question:

Is it possible to make a tour so that one passes just once over all the bridges over the river Preger in Königsberg?

Euler found it impossible to make such a tour. The arrangement of the bridges according to Kraitchik (Mathematical Recreations, Dover, 1958) is given in the figure below.

Encouraged by the other recent advances in mathematics, I decided to visit Königsberg (nowadays Kaliningrad) and have a go at the problem. It turned out I was able to make a tour crossing all the bridges only once over the river Preger in Königsberg!

My solution: Start from D, pass to B over ff. Then go from B to C passing over aa and cc (in fact aa and cc are built as a single bridge passing above A with a stairway in the middle leading down to A). Then back to D over gg and further to A over ee. In order to include the stairway as well, climb the stairs from A up to aa-cc.

The solution was possible in spite of the fact that the bridges bb and dd are no longer there, thus limiting the number of move options for the problem solver.

## Photo proofs:

Pointing out bridge a-a from point B.

Stopping at the bridge e-e on the way to A. The church in the background is on island A.

Bridge e-e from A. Bridge e-e was originally constructed to be lifted. Thankfully, today it's in no condition to be lifted. Otherwise, the tour problem with bridge e-e lifted would have presented a new challenge.

Bridge g-g seen from island A.

Driving over bridge g-g towards shore C. Note the building to the left, next picture.

A big but abandoned building at C. Could be the remains of a gigantic effort to solve the ancient geometric problem of doubling the cube.

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