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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.
Contact: jankahre (at) hotmail.com