WEBVTT

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

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but one particular quirk in its geography

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

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

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At the center were two large islands.

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

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by seven bridges.

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

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grew obsessed with these islands
and bridges.

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He kept coming back to a single question:

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Which route would allow someone
to cross all seven bridges

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without crossing any of them
more than once?

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Think about it for a moment.

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7

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6

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5

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4

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3

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2

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1

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Give up?

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You should.

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It's not possible.

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But attempting to explain why
led famous mathematician Leonhard Euler

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to invent a new field of mathematics.

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Carl wrote to Euler for help
with the problem.

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

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But the more he wrestled with it,

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

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

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

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now known as Graph Theory.

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Euler's first insight

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

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didn't actually matter.

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

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represented as a single point,

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what we now call a node,

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

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

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

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Why do the degrees matter?

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Well, according to the rules 
of the challenge,

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once travelers arrive onto a landmass
by one bridge,

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they would have to leave it 
via a different bridge.

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

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must occur in distinct pairs,

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meaning that the number of bridges
touching each landmass visited

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must be even.

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The only possible exceptions would be
the locations of the beginning

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and end of the walk.

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

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So no matter which path is chosen,

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

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Euler used this proof to formulate
a general theory

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that applies to all graphs with two
or more nodes.

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A Eulerian path 
that visits each edge only once

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is only possible in one of two scenarios.

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

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meaning all the rest are even.

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

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and the end point is the other.

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

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

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which also makes it something called
a Eulerian circuit.

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So how might you create a Eulerian path
in Königsberg?

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It's simple.

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Just remove any one bridge.

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

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

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making a Eulerian path easily possible.

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

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These bombings pretty much wiped
Königsberg off the map,

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

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So while Königsberg and her seven bridges
may not be around anymore,

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they will be remembered throughout
history by the seemingly trivial riddle

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