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You'd have a hard time finding[br]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[br]in mathematics.

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

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At the center were two large islands.

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The two islands were connected [br]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[br]later became the mayor of a nearby town,

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grew obsessed with these islands[br]and bridges.

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He kept coming back to a single question:

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Which route would allow someone[br]to cross all seven bridges

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without crossing any of them[br]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[br]lead 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[br]with the problem.

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Euler first dismissed the question as[br]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[br]be something there after all.

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

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that did not quite exist yet,[br]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[br]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[br]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[br]to represent the bridges.

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

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

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Why do the degrees matter?

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Well, according to the rules [br]of the challenge,

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once travelers arrive onto a landmass[br]by one bridge,

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they would have to leave it [br]via a different bridge.

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In other words, the bridges leading[br]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[br]touching each landmass visited

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must be even.

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The only possible exceptions would be[br]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[br]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,[br]a bridge will have to be crossed twice.

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Euler used this proof to formulate[br]a general theory

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that applies to all graphs with two[br]or more nodes.

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A Eulerian path [br]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[br]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[br]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[br]are of even degree.

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

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which also makes it something called[br]a Eulerian circuit.

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So how might you create a Eulerian path[br]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[br]a Eulerian path of its own.

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During World War II, the Soviet Air Force[br]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[br]wasn't their intention.

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These bombings pretty much wiped[br]Königsberg off the map,

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

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

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they will be remembered throughout[br]history by the seemingly trivial riddle

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