Königsberg

Can the Bridge of Königsberg problem be solved? Explain your reasons with the aid of illustration(s).

Introduction The **Seven Bridges of Königsberg** is a famous historical problem in mathematics. Its negative resolution by Leonhard Uler in 1735 laid the foundations of graph theory and presaged the idea of topology

Solution to The Bridges of Konigsberg
As the great German mathematician Leonhard Euler showed, any tour that crossed each of the bridges once and only once was equivalent to a tracing of the graph below in which your pencil must trace each line just once. As you can see, all four points lie on an odd number of lines. Thus, even if you start and end your tracing at different points, you must have gone through the other two points an even number of times and this is impossible if you traced each of the edges in the graph. For these two points also have an odd number of lines entering/leaving them.

-  [|Euler] approached this problem by collapsing areas of land separated by the river into points, which he labelled with capital letters. Modern graph theorists call these //vertices//, and have gone on to represent them and bridges graphically. For Konigsberg, let us represent land with red dots and bridges with black curves, or //arcs//:  Thus, in its stripped down version, the seven bridges problem looks like this: 

 The problem now becomes one of drawing this picture without retracing any line and without picking your pencil up off the paper. Consider this: all four of the vertices in the above picture have an odd number of arcs connected to them. Take one of these vertices, say one of the ones with three arcs connected to it. Say you're going along, trying to trace the above figure out without picking up your pencil. The first time you get to this vertex, you can leave by another arc. But the next time you arrive, you can't. So you'd better be through drawing the figure when you get there! Alternatively, you could start at that vertex, and then arrive and leave later. But then you can't come back. Thus every vertex with an odd number of arcs attached to it has to be either the beginning or the end of your pencil-path. So you can only have up to two 'odd' vertices! Thus it is impossible to draw the above picture in one pencil stroke without retracing.

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