The Seven Bridges of Konigsberg

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It is now called Kaliningrad

Can you see the seven bridges?

The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

Problem 1Try it. Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke

Can’t do it - neither could Euler - a very famous mathematician. In fact he proved that it couldn’t be done.

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Leonhard Euler

Problem 2Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:

Now try and solve the problem

Problem 3Does it matter which bridge you take away? What if you add bridges? Come up with some maps on your own, and try to 'plan your journey' for each one

Can you draw any conclusions?

These are the same Diagram

Node(Vertice)Edge

(Arc)

A node is ODD if it has an odd number of edges leading into it otherwise it is called

even

A network is a figure made up of nodes and edges

An Euler path is a continuous path that passes through each arc once and only once

- we say the network is transversable

Euler proved:

If a network has more than two odd vertices, it does not have an Euler path i.e it is not transversable

If a network has two or zero odd vertices, it has at least one Euler path. In particular, if a network has exactly two odd vertices, then its Euler path can only start on one of the odd vertices

He also proved:

Why is this important.....Circuits?

This branch of Mathematics is called Graph Theory

or more specifically topological graph theory

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Very useful for proving the “Hairy Ball Theorem”

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Just to alter your perception of reality a bit...