In the 1700s, Leonard Euler was intrigued by a question in the city of Königsberg, which was made up four main pieces of land separated by a river and connected by seven bridges. Walking the city was a popular leisure activity the question arose as to whether it was possible to walk the city and cross each bridge once, and only once.
- Here is a basic sketch of the situation. Find a path someone could take that crosses each bridge exactly one time:
- After some incredible frustration, and attempting every single path, you have probably come to the conclusion that this is impossible. But now you are given the ability to take away a bridge. Is it possible now? Does it matter which bridge you take away? Does your starting point matter?
- Attempt problem two again, but this time you can add a bridge anywhere, for a total of eight.
- There is no reason to stop at this simple map. Is it possible to walk the city below, while still restricted to crossing each bridge exactly once?
- Euler went further than simply testing every possibility. He invented a technique for quickly assessing whether or not a given configuration is possible. Can you invent a strategy?
- Look up Euler’s proof and his notation for describing a region simply. In your own words, explain his proof and technique.
- Create some bridge/land configurations of your own, apply Euler’s graph theory, and prove/disprove that his proof is actually correct.