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  • You'd have a hard time findingnigsberg on any modern maps,

  • but one particular quirk in its geography

  • has made it one of the most famous cities in mathematics.

  • The medieval German city lay on both sides of the Pregel River.

  • At the center were two large islands.

  • The two islands were connected to each other and to the river banks

  • by seven bridges.

  • Carl Gottlieb Ehler, a mathematician who later became the mayor of a nearby town,

  • grew obsessed with these islands and bridges.

  • He kept coming back to a single question:

  • Which route would allow someone to cross all seven bridges

  • without crossing any of them more than once?

  • Think about it for a moment.

  • 7

  • 6

  • 5

  • 4

  • 3

  • 2

  • 1

  • Give up?

  • You should.

  • It's not possible.

  • But attempting to explain why led famous mathematician Leonhard Euler

  • to invent a new field of mathematics.

  • Carl wrote to Euler for help with the problem.

  • Euler first dismissed the question as having nothing to do with math.

  • But the more he wrestled with it,

  • the more it seemed there might be something there after all.

  • The answer he came up with had to do with a type of geometry

  • that did not quite exist yet, what he called the Geometry of Position,

  • now known as Graph Theory.

  • Euler's first insight

  • was that the route taken between entering an island or a riverbank and leaving it

  • didn't actually matter.

  • Thus, the map could be simplified with each of the four landmasses

  • represented as a single point,

  • what we now call a node,

  • with lines, or edges, between them to represent the bridges.

  • And this simplified graph allows us to easily count the degrees of each node.

  • That's the number of bridges each land mass touches.

  • Why do the degrees matter?

  • Well, according to the rules of the challenge,

  • once travelers arrive onto a landmass by one bridge,

  • they would have to leave it via a different bridge.

  • In other words, the bridges leading to and from each node on any route

  • must occur in distinct pairs,

  • meaning that the number of bridges touching each landmass visited

  • must be even.

  • The only possible exceptions would be the locations of the beginning

  • and end of the walk.

  • Looking at the graph, it becomes apparent that all four nodes have an odd degree.

  • So no matter which path is chosen,

  • at some point, a bridge will have to be crossed twice.

  • Euler used this proof to formulate a general theory

  • that applies to all graphs with two or more nodes.

  • A Eulerian path that visits each edge only once

  • is only possible in one of two scenarios.

  • The first is when there are exactly two nodes of odd degree,

  • meaning all the rest are even.

  • There, the starting point is one of the odd nodes,

  • and the end point is the other.

  • The second is when all the nodes are of even degree.

  • Then, the Eulerian path will start and stop in the same location,

  • which also makes it something called a Eulerian circuit.

  • So how might you create a Eulerian path innigsberg?

  • It's simple.

  • Just remove any one bridge.

  • And it turns out, history created a Eulerian path of its own.

  • During World War II, the Soviet Air Force destroyed two of the city's bridges,

  • making a Eulerian path easily possible.

  • Though, to be fair, that probably wasn't their intention.

  • These bombings pretty much wipednigsberg off the map,

  • and it was later rebuilt as the Russian city of Kaliningrad.

  • So whilenigsberg and her seven bridges may not be around anymore,

  • they will be remembered throughout history by the seemingly trivial riddle

  • which led to the emergence of a whole new field of mathematics.

You'd have a hard time findingnigsberg on any modern maps,

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B1 TED-Ed euler path mathematics node graph

【TED-Ed】How the Königsberg bridge problem changed mathematics - Dan Van der Vieren

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    小爸 posted on 2017/07/04
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