Konisberg Bridges ( One-way street ) SOL: DM.2

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

2. Senior Exemption from Final Exam

4. 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 18th 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…

5. 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!

6. How many bridges are there? 4 How many cities are there? 7

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

10. Can you see a pattern? More than two odd vertices, then you cannot trace it successfully

11. 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?

12. The Konigsberg Problem • Is it traversable? Justify. • YES, no more than 2 odd vertices • 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 greenare ‘odd’ • Fill in the table, see if you can find a pattern that allows you to tell whether a network is traversable or not! O O E E E E E E E

15. Plenary Network Traversable? Even vertices Odd vertices Number • 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? 1 Y 1 2 2 Y 2 2 3 N 1 4 4 Y 3 2 5 N 0 4 6 N 0 8 7 Y 5 0 8 Y 7 2 9 Y 6 0 10 Y 8 0 11 Y 8 0 12 Y 11 0 13 N 0 4 14 N 2 4 15 Y 4 2 16 N 2 4 17 Y 4 0

16. So….is the Konigsberg Bridge Traversable? Justify. No, because 0 even vertices and 4 odd vertices The Konigsberg problem This is the map of the Konigsberg bridges, turned into a network Konigsberg This result (and the rule you have seen) was first proved by Leonard Euler in 1735!

17. The underground map – easy to follow! 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 actual underground layout – seems more complicated!!