One day a guy approached us with a puzzle he has been pondering on for approximately seven years. He called this the ‘Walls and Lines Puzzle’.

Rules and Regulations Each vertex is considered a ‘Node’. From node to node is considered a ‘Bridge’. You must cross each bridge once and only once with one continuous line. You may cross YOUR line. You CANNOT run you line through a node.

Here is an example of the many ways we have tried to solve this problem. As you can see we crossed all but one line, therefore this is not a solution.

Seven Bridges of Konigsberg In Konigsberg, Prussia (present day Kaliningrad, Russia) the river Pregel divides the town into four separate land masses-A, B, C, & D. Some of the citizens wanted to see if they could cross all seven bridges without crossing one more than once. Many tried and everyone failed.

Leonhard Euler’s Answer Around the mid-1730’s a man named Leonhard Euler came up with the answer. –Is it possible to cross all seven bridges once and only once? NO!!! It is impossible and he came up with a reasoning we soon adopt as Graph Theory.

Euler’s Reasoning –Anyone standing on a land mass would have to have a way to get on and off. –So, each land mass would need and even number of bridges connecting to it.

Let’s start out with a single node with an odd number of bridges. If this is NOT your starting point then you would: –Enter –Leave –Enter Again Thus, if you start somewhere else-with this having an odd number of bridges you realize that you must END here.

Now, let’s add another point with an odd number of bridges. Therefore you must start at one point and end at the other. If we start at ‘Home’ we must end at point B. The pathway would be –Leave Home –Enter B –Leave B –Enter Home –Leave Home –Enter B

So far we have concluded that if a point has an odd number of bridges then we will sooner or later end up leaving it and not being able to reenter. With two nodes having odd bridges we can leave one and finish by ending on the other. Therefore you must start and end on a point with an odd number of bridges. Recap

Now, let’s add a third point with an odd number of bridges. We already know that you must start and end on a point with an odd number of bridges. That leads us to our obstacle –There are three points with an odd number of bridges. –And we can only start at 1 point and end at 1 point. –Therefore either a third bridge is going to have to be removed or this is impossible to solve.

You have your four different land masses- A, B, C & D. They can be represented by ‘Nodes’. The bridges/pathways can be represented by lines/arcs. Now let’s reconstruct Euler’s bridge diagram.

Since we are starting at one node and ending at another only 2 nodes can have an odd number of bridges. We will classify each node as either odd or even depending on its number of bridges. Node A- 5 Bridges=Odd Node B- 3 Bridges=Odd Node C- 3 Bridges=Odd Node D- 3 Bridges=Odd So since out of the 4 nodes more than two of them are odd it is IMPOSSIBLE to cross all bridges once and only once!!!

Let’s Apply that to the first problem we gave you! Let’s circle all the “odd nodes”. Since there are more than 2 “odd nodes” it is impossible to cross each line once and only once!!!

Extra Information This whole idea of Graph Theory arose because of recreation. People in the 1700’s wanted something to do and they worked on this for a long time. This was nothing more than Pure Mathematics and now we use it in everyday life. Graph Theory is used today in things like, electrical engineering, computer programming and networking. Anything that deals with pathways can be related to Graph Theory. For instance –Mail-routes –Snow Removal –Garbage Pick-ups –Even programming stop lights in a town

Presented by: Shelby Durler & Emily Tenbrink