1. Konisberg Bridges (One-way street) SOL: DM.2 Classwork Quiz/worksheet Homework (day 62) worksheet

2. http://www.mathsisfun.com/activity/seven-bridges-konigsberg.html

7. Seven Bridges of Konigsberg https://www.youtube.com/watch?v=Kw6g31HFMDA

9. Starter Two puzzles for you to start with: 1)Can you draw the shape to the right without taking your pen off the page, and without crossing the same line twice? Can you do it from any starting position? 2)Can you draw 4 straight lines without taking your pen off the page, that go through all the dots in the grid to the right?

10. The Konigsberg Problem Konigsberg was a city in Prussia (it is now called Kaliningrad and is in Russia) It gives its name to a mathematical problem of the 18 th century Konigsberg To the right is a drawing of Konigsberg. The problem was as follows: “Is it possible to walk around the city, crossing each bridge once and only once?” The bridges are now shown in red…

11. The Konigsberg Problem To answer this problem we are going to look into the mathematics of networks A ‘network’ in Maths is similar to a map It is made up of vertices (dots) and arcs (lines joining the dots) Some networks are traversable – this means you can travel round the whole network, using each arc once only Our objective is to see if we can prove whether the Konigsberg problem is solvable or not! Start by looking at the networks on the sheet you have been given. Pick various starting points and try to decide whether each network is traversable or not… The above network (from the starter) IS traversable as you can travel along every arc without overlapping any!

12. The Konigsberg Problem This is network 8, it is traversable The vertices can be described as even or odd  An even vertex has an even number of arcs joined to it  The vertices marked in blue are ‘even’  An odd vertex has an odd number of arcs joined to it  The vertices marked in green are ‘odd’ Fill in the table, see if you can find a pattern that allows you to tell whether a network is traversable or not! E E E E E E E O O

13. Plenary Do you notice any patterns in the networks that are traversable?  If a network is to be traversable, it will always have 0 or 2 ‘odd’ vertices…  But can you explain why this is? Network Number Traversable?Even verticesOdd vertices 1Y12 2Y22 3N14 4Y32 5N04 6N08 7Y50 8Y72 9Y60 10Y80 11Y80 12Y110 13N04 14N24 15Y42 16N24 17Y40

14. Plenary Imagine we create a network of our own… Odd  The initial vertex will have an odd number of arcs joining it (1)  After this, every vertex visited will have an entrance point and exit point, so will have an even number of arcs… 1 24 24 6 2 2  At the end, we can either finish by joining to an even vertex, giving us 2 odd vertices  Or, we finish by joining to the original starting vertex, making them all even and hence having no odd ones at all! ??? Even 3Odd 2

15. Plenary The Konigsberg problem Konigsberg This is the map of the Konigsberg bridges, turned into a network Even vertices  0 Odd vertices  4 So the network is NOT traversable! This result (and the rule you have seen) was first proved by Leonard Euler in 1735!

16. Summary We have learnt about a famous mathematical problem based on the city of Konigsberg This problem led into a new branch of mathematics called topology One of the key ideas of topology is that it does not matter how some things are arranged, what matters is how they are connected A great example is the London underground map. People aren’t that bothered exactly where the stations are, they care about how to get from one station to another! The underground map – easy to follow! The actual underground layout – seems more complicated!!