In 1736, Königsberg had seven bridges. The puzzle: walk through the city crossing each bridge exactly once. Euler proved it's impossible — and invented graph theory in the process.
Quick check
Königsberg had 7 bridges joining 4 land masses, and every land mass touched an ODD number of bridges. Can you take a walk that crosses each bridge exactly once?
Euler's bridge rule
- Turn the map into a graph: land = vertices, bridges = edges.
- Count each vertex's degree (how many edges touch it).
- 0 odd-degree vertices → an Eulerian *circuit* exists (start = end anywhere).
- Exactly 2 odd-degree vertices → an Eulerian *path* exists (must start at one odd vertex, end at the other).
- More than 2 odd vertices → impossible. (Königsberg had 4.)
Your turn
Can you draw a standard envelope shape (rectangle with an X-roof on top) in one stroke without lifting your pen?
The rule: a graph has an Eulerian path (one that crosses each edge once) only if it has exactly 0 or 2 vertices with an odd number of edges. Königsberg had 4 — so no path exists.