[Leonhard Euler](http://en.wikipedia.org/wiki/Leonhard_Euler) proved that it is impossible to solve [this puzzle](http://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg):

![wikimedia](https://upload.wikimedia.org/wikipedia/commons/thumb/9/91/7_bridges.svg/500px-7_bridges.svg.png)

The challenge is to take a walk around the area depicted, and cross each bridge (yellow) *exactly* once. Using topology, Euler proved that it is impossible to walk each bridge exactly once. 

Are there any similar challenges? I would like to try to solve a more complex one.