These puzzles are named after Leonhard Euler who was a brilliant 18th century mathematician who solved the puzzle of the Konigsberg bridges (now Kaliningrad, in Russia). Seven bridges connected both sides of the city and two islands in between. The townsfolk passed their time for many years trying (unsuccessfully) to find a route which would cross each bridge only once. Euler proved that is was not possible, until an eighth bridge was constructed.
Here is Konigsberg with its bridges shown. Imagine walking around them!
Here is a schematic that can help solving this problem. Note how we are representing the problem better in terms of ‘essentials’.
And here is the equivalent represented as a ‘topological map’ or graph which represents just the ‘nodes’ and ‘links’. (The London Underground map is an example of a topographical map). This is now looking like the kind of Euler puzzle you have just done.
In this graph you can see there are 4 areas of land (the nodes) and 7 bridges(the links or edges), and by tracing out different paths you can quickly see that there is no solution to this problem for this graph. Topological maps make representing the problem much easier. They are not geographically accurate, but let us see the ‘state space’ of the situation better – and you can quickly trace out the different possibilities.
Mindware strategy tips
In solving this kind of problem, one tip is obviously to ensure you represent the problem in a good ‘graph’. Euler did this in solving his bridge problem. Another tip is to use ‘means-ends’ subgoal analysis. For instance ‘ending up at the same point’ for half of the figure, and then repeating the path for the other half. If you can extract any ‘general rules’ for this kind of problem, this can help you when you come across other problems too.
Additional resources
An explanation of Euler’s rule allowing you to create your own Euler puzzles at will!
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