## Subtitles section Play video

• 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,

Subtitles and vocabulary

Operation of videos Adjust the video here to display the subtitles

B1 TED-Ed euler path mathematics node graph

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

• 319 26
小爸 posted on 2017/07/04
Video vocabulary