Königsberg Bridges

Attempt the walk on the actual map before simplifying anything.

1. Draw the two banks, two islands and seven bridges.
2. Trace routes with a pencil, trying to cross every bridge exactly once.
3. Record how many attempts you make and where each one fails.

Reduce the map to dots and lines.

1. Replace each landmass with a single dot — four dots.
2. Replace each bridge with a line joining two dots — seven lines.
3. Redraw it several times with the dots in different places.

One count answers the question.

1. For each dot, count the lines meeting it — its degree.
2. Mark which dots have an ODD degree.
3. In Königsberg, all four are odd.

The same count tells you when a route IS possible, which is the useful half.

1. Redraw the graph, adding one bridge to make exactly two odd vertices.
2. Find a walk — it must start at one odd vertex and end at the other.
3. Now make all vertices even and find a walk that returns to its start.

Leonhard Euler presented the solution to the St Petersburg Academy in 1735 and published in 1736. He was mildly dismissive of the problem itself — it looked like a triviality with no mathematics in it — and what he actually contributed was the method of turning it into a question about connection alone.

The city has changed and the puzzle with it. Königsberg is now Kaliningrad; two bridges were destroyed in the Second World War and others rebuilt, and with the modern arrangement a walk is possible. The mathematics did not change — the graph did.

What the approach opened up: graph theory now underlies routing on networks, scheduling, dependency resolution in software builds, molecular structure in chemistry, and social network analysis. The Chinese postman problem — find the shortest route covering every edge, repeating as few as necessary — is the practical descendant used to plan refuse collection and gritting rounds.

The transferable idea is the discarding. Euler's contribution was recognising which features of a situation carry the answer and which are decoration. That judgement is doing the same work in every abstraction since: a circuit diagram ignores the physical layout of wires, an underground map ignores real distances, and both are more useful for it.