For the next few posts I’m going to post about Maths and the world cup , and from some of the countries competing. Today its not on one on the countries, but one of the cities where matches are taking place.
Kalilingrad, where England play Belgium is in a part of Russia now separate from the rest of the country but it used to be a German City called Konigsberg. The city is built on the River Pregel, which flows through the city leaving a number of islands. In the 18th Century there were 7 bridges across the river, like this
The citizens of Konigsberg liked to challenge visitors with the following task – Can you walk around the city, starting and finishing at the same point and cross each bridge only once. No body was able to do this, but neither could anybody show that it was impossible.
The the Swiss mathematician Euler got involved – and more of him when I get to Switzerland. He showed that by simplifying the map to just dots and lines, it could be shown that it was impossible, and with that started a whole new branch of mathematics called Network Theory, which later became part of the whole new area of Topology. All because the people of Konigsberg liked to challenge their visitors
What Euler did first was that Euler draw the map as lines and dots, removing the ‘Cityness’ of the map. This is a normal thing to do in ‘mathematical Modelling’ – Strip what he have down to the basics
I won’t use all his mathematical language here but you might see the sense of what he said. Each point has a number of lines and Euler called this the ‘order’. He showed that to be sure of being able to do the walk, all the points needed an even number of lines. He also showed that you can have two points with an odd number, just so long as you didn’t want to get back to where you started (and started at one of the ‘odd’ number points.
Why not draw some of your own diagrams and check this rule out?