Taking a Walk in Venice
I was going for a walk in Venice, crossing over the canals via dozens of bridges, and it made me think of a famous famous problem in mathematics called the Seven Bridges of Königsberg. It was about the city of Königsberg, which was in Prussia but is now located in Russia. A river ran through the city, and there were two large islands in the river. The islands and the riverbanks were connected by seven bridges, and the question was whether or not it was possible to go for a walk crossing each bridge exactly once. Take a look at the maps below (the left is an image from Google Maps, and the right is a simpler drawing of the situation) and see if you can create a path that crosses each bridge exactly once:
Are you ready for the answer? It is actually not possible to do this. The simplest way to think about this is to look at each piece of land and the number of bridges attached to it:
If a piece of land has an odd number of bridges attached to it, then it must be either the starting point of the walk or the ending point of the walk. Because if it's not a starting or ending point, then you have to cross one bridge when you arrive on the piece of land and a second bridge when you leave. So that's two bridges. And if you cross over that piece of land twice, then that's four bridges (two arrivals and two departures).
Because you can only start in one place and end in one place, you can have at most two pieces of land connected to an odd number of bridges. Those would then be your starting and ending points.
If you look at the map here, you can see that there are four pieces of land connected to odd numbers of bridges. Because of this, it is not possible to take such a walk in Königsberg. This is just a single problem, but there's a whole branch of mathematics called graph theory that deals with these sorts of things.
So then, what about Venice? The main part of the city is made of over a hundred islands, and certainly there are more than two islands connected by an odd number of bridges. So it can't be possible. However, I took a day trip to the islands of Murano and Burano, and as I was going for a wander, I wondered if such a walk would be possible. Take a look at the maps and decide for yourself (I have highlighted the bridges in black to make it easier to see them all).
Sample Problems
1. Take a look at the islands of Murano. Is it possible to take a walk that crosses each bridge exactly once? If it is possible, determine the path you could take. If it is not possible, explain why it is not.
2. Take a look at the islands of Burano. Is it possible to take a walk that crosses each bridge exactly once? If it is possible, determine the path you could take. If it is not possible, explain why it is not.
3. Use Google Maps to find a chain of islands somewhere else in the world and determine whether or not it is possible to take a walk that crosses each bridge exactly once.
