# Networks - Konigsberg Bridge Problem.

• Shan Goonewardena
• May 25, 2008
• Networks

In the country of Germany, the river Pregel divides the town of Kaliningrad into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and the problems is if it is possible to take a journey across all seven bridges without having to cross any bridge more than once, mathematical we are trying to find if this network is traversable or not. On the right side is a diagram of the situation and the dotted letters are the four zones, mathematically referred as nodes (I will talk about this later on). To overcome this problem, I not only going to find the solution to this problem but also state a rule, which will tell if a network is traversable or not.

The worksheet helped me a lot in solving this problem and basically the worksheet consisted of three parts. Firstly, the worksheet provides a list of networks and we asked to find if the networks as traversable or not. With these results, we had to create a table with the number of even and odd nodes, which is very important. A Node is a point in a network at which lines intersect or branch.

For example: This network has nine nodes (the dots) but

it has 5 even nodes and 4 odd nodes, because for each

node, there is a different number of lines intersecting. When we find the number lines branching out of each

node, we count all the even numbers as 1 so in this case, 5

even nodes and the same for the odd nodes. Now that you

know a thing about nodes, let me continue with the explanation of  worksheet.

After you fill in the table, we are left to find a pattern in the table and the importance of even and odd nodes are very essential at that stage. Then in the second part, we test out our rule or check if we have to start with an even or odd node. All these will be better explained when I tell what the rule is. And finally in the third section, there is another map of the bridges in Kaliningrad except with two more bridges across the Pregel. For this part, I have to apply my rule and tell if the network is traversable.

For the first part, the table was the most beneficial but the networks provided the information, which was later added to my table. I didn’t know the rule at first so to check if the network is traversable, I drew the whole network in one movement, without lifting the writing utensil and also without going over the same part twice. At first, I didn’t get the point of the nodes but I was wrong, the nodes are pretty much the focal point of the rule.

Below are the networks, which were asked if ...