Seven Bridges

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.

  1. Here is a basic sketch of the situation. Find a path someone could take that crosses each bridge exactly one time:Konig Basic
  2. 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?
  3. Attempt problem two again, but this time you can add a bridge anywhere, for a total of eight.
  4. 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?Konig Adv
  5. 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?
  6. Look up Euler’s proof and his notation for describing a region simply. In your own words, explain his proof and technique.
  7. Create some bridge/land configurations of your own, apply Euler’s graph theory, and prove/disprove that his proof is actually correct.