1. The Seven Bridges
of Konigsberg
QuickTime™ and a
H.264 decompressor
are needed to see this picture.

2. It is now called Kaliningrad
Can you see the seven bridges?

4. The people wondered whether or not one could walk around the city in a way that
would involve crossing each bridge exactly once.
Problem 1
Try 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

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

6. Leonhard Euler
QuickTime™ and a
decompressor
are needed to see this picture.

7. Problem 2
Suppose they had decided to build one fewer bridge in Konigsberg, so
that the map looked like this:
Now try and solve the problem

8. Problem 3
Does 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?

9. These are the same
Diagram

10. Node
(Vertice)
Edge
(Arc)
A network is a figure made up of nodes and edges
A node is ODD if it has an odd number of edges
leading into it otherwise it is called even
An Euler path is a continuous path that passes
through each arc once and only once - we say the
network is transversable

11. Euler proved:
If a network has more than two odd vertices,
it does not have an Euler path i.e it is not
transversable
He also proved:
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
Why is this important..... Circuits?

12. This branch of Mathematics is called Graph Theory
or more specifically topological graph theory
Very useful for proving the “Hairy Ball Theorem”
QuickTime™ and a
decompressor
are needed to see this picture.

13. Just to alter your perception of reality a bit...
QuickTime™ and a
decompressor
are needed to see this picture.