Bridges Of KoenigsbergYou are currentlynot logged in Click here to log in 

Each Sunday people would go for a walk. After a while someone noticed that no matter where they started they were never able to cross each bridge once and only once during their walk. It seemed that no matter where they started and where they walked, they would always end up in the wrong place for crossing that last bridge.
Try it for yourself.
It was suggested that perhaps it was impossible, but there was always the nagging doubt that perhaps it was possible and people simply hadn't (yet!) found how to do it.
Then along came a clever chap called Leonhard Euler (pronounced "Oiler") who settled the matter once and for all. In doing so he used the main ideas found in all mathematics:
Because walking around on land or islands was irrelevant, he shrunk each of them to a single point, leaving only the lines representing the bridges.
Now observe that if you can draw this diagram without lifting your pen and without going over any line twice, that gives you your Sunday walk.
Since X has some bridges coming to it we will have to visit it at least once. Since we neither start nor finish there, every time we come in we must go out again, and on a different bridge. That means that the total number of bridges that land at X must be even. For every in there is an out, and if there's an odd number of bridges, that won't work.
So X, our place that is neither start nor finish, must have an even number of bridges. However, all our landing places have odd numbers of bridges,and that makes it impossible to do our walk.
(none)  (none)  GraphTheory  
(none)  FourColourTheorem  
⇌ 
You are hereBridgesOfKoenigsberg 

Euler  
(none)  (none)  Calculus FamousPeople GraphTheory 
Last change to this page Full Page history Links to this page 
Edit this page (with sufficient authority) Change password 
Recent changes All pages Search 