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Königsberg Bridges
Mark

सिर्जनाकर्ता

Mark

20. अगस्ट 2026FI
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Königsberg Bridges

The city of Königsberg sat on two banks and two islands of the river Pregel, joined by seven bridges, and the townspeople had a standing puzzle: find a walk that crosses every bridge exactly once. Euler settled it in 1736 by discarding almost everything about the map. Distances do not matter, shapes do not matter, and the positions of the bridges do not matter — only which landmasses connect to which, and how many bridges meet at each. Reduced to that, the answer falls out of a simple count of odd connections, and no such walk exists. The paper is generally taken as the beginning of graph theory and of topology, because it is the first time a problem was solved by deliberately throwing geometry away.
शुरुआती
45 minutes

निर्देशनहरू

1

Try it the obvious way first

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.
You will keep stranding yourself with one bridge unused. Note WHERE the failures happen — the pattern in the failures is the evidence Euler generalised, and it is worth collecting before being told the answer.

Materials for this step:

Graph PaperGraph Paper1 pad
Graphite Pencil SetGraphite Pencil Set1 सेट
2

Throw the geography away

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.
Every version is the same problem. Distances, angles and the river have all vanished and nothing was lost, because the walk only ever depended on what connects to what. Deciding what to discard is the actual move here.
3

Count the odd vertices

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.
Here is the rule: except at the start and end, every visit to a landmass uses one bridge to arrive and one to leave, in pairs. So a landmass with an odd number of bridges must be either the start or the end. With four odd vertices you would need four endpoints, and a walk has two. No such walk exists — and this is a proof, not a failure to find one.
4

Find where the approach does apply

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.
Zero odd vertices gives a closed circuit; exactly two gives an open path between them; anything else gives nothing. Postal rounds, street-sweeping routes, snow ploughing and circuit-board drilling are all planned with this and its descendants.

Materials for this step:

Cardstock Assorted Pack (50 Sheets)Cardstock Assorted Pack (50 Sheets)1 प्याक
5

History and context

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.

सामग्री

3

CC0 सार्वजनिक डोमेन

यो ब्लुप्रिन्ट CC0 अन्तर्गत जारी गरिएको छ। तपाईं अनुमति नसोधी प्रतिलिपि, परिमार्जन, वितरण र प्रयोग गर्न सक्नुहुन्छ।

ब्लुप्रिन्ट मार्फत उत्पादनहरू किनेर सिर्जनाकर्तालाई सहयोग गर्नुहोस् सिर्जनाकर्ता कमिसन विक्रेताले तोकेको, वा यो ब्लुप्रिन्टको नयाँ संस्करण बनाउनुहोस् र आम्दानी बाँड्न आफ्नो ब्लुप्रिन्टमा जडानको रूपमा समावेश गर्नुहोस्।

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