Konigsburg Bridge Problem

• The Konigsberg Bridge problem is a famous mathematical problem studied by many students in geometry.

• Its citizens pondered for a long time whether it was possible to walk about the city in such a way that you cross all seven bridges (yellow in diagram) exactly once.

• Euler trimmed the problem down to its basics.

• The various islands and pieces of land became dots

• The paths between the pieces of land became line segments connecting the dots

• Can you start at one of the pieces of land and travel all seven bridges exactly once?

• Euler proved that it was impossible to make such a walk.

• How did he do it?• Let’s explore various

networks to see if we can see that Euler discovered?

• First let’s define an even point and odd point.

An even point

An odd point

• Next you will see a series of networks. Study they in several ways. – Can you travel the networks? Record your

answer.– How many odd points make up the network?– How many even points make up the network?– How many total points make up the network?

• Complete the chart.

Some Questions to Ponder

• Of the networks that could be traveled, how many of the networks had an odd number of odd points?

• Of the networks that could be traveled, how many of the networks had an even number of odd points?

• Of the networks that could be traveled, how many of the networks had an odd number of even points?

• Of the networks that could be traveled, how many of the networks had an even number of even points?

• Shade in the columns on the chart for all networks that can be traveled.

• Look to see how many odd points a network must have to be traveled. Does it seem to matter how many even points a network contains?

• What do you notice about the number of odd points on a network that can be traveled? (How many odd points must it have?)

Euler’s ObservationEuler’s ObservationEuler’s ObservationEuler’s Observation

What must be true about the What must be true about the number of odd points in a number of odd points in a

network if it can be traveled?network if it can be traveled?

Using Euler’s Conjecture

• Study Konigsberg Bridge diagram Notice the drawing has 4 odd points. Use your conjecture to add a path so the number of odd points will make it possible to travel. Describe how this addition fits your conjecture.

Showing your Understanding

Make two new networks up: one that cannot be traveled and one that can be traveled. Use your conjecture to explain why each can or cannot be traveled

A NETWORK THAT CAN BE TRAVELED

A NETWORK THAT CANNOT BE TRAVELED