[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:09.04,0:00:14.11,Default,,0000,0000,0000,,You'd have a hard time finding\NKönigsberg on any modern maps,
Dialogue: 0,0:00:14.11,0:00:17.42,Default,,0000,0000,0000,,but one particular quirk in its geography
Dialogue: 0,0:00:17.42,0:00:22.20,Default,,0000,0000,0000,,has made it one of the most famous cities\Nin mathematics.
Dialogue: 0,0:00:22.20,0:00:26.21,Default,,0000,0000,0000,,The medieval German city lay on both sides\Nof the Pregel River.
Dialogue: 0,0:00:26.21,0:00:28.88,Default,,0000,0000,0000,,At the center were two large islands.
Dialogue: 0,0:00:28.88,0:00:33.12,Default,,0000,0000,0000,,The two islands were connected \Nto each other and to the river banks
Dialogue: 0,0:00:33.12,0:00:35.88,Default,,0000,0000,0000,,by seven bridges.
Dialogue: 0,0:00:35.88,0:00:41.30,Default,,0000,0000,0000,,Carl Gottlieb Ehler, a mathematician who\Nlater became the mayor of a nearby town,
Dialogue: 0,0:00:41.30,0:00:44.40,Default,,0000,0000,0000,,grew obsessed with these islands\Nand bridges.
Dialogue: 0,0:00:44.40,0:00:47.20,Default,,0000,0000,0000,,He kept coming back to a single question:
Dialogue: 0,0:00:47.20,0:00:51.10,Default,,0000,0000,0000,,Which route would allow someone\Nto cross all seven bridges
Dialogue: 0,0:00:51.10,0:00:55.14,Default,,0000,0000,0000,,without crossing any of them\Nmore than once?
Dialogue: 0,0:00:55.14,0:00:56.95,Default,,0000,0000,0000,,Think about it for a moment.
Dialogue: 0,0:00:56.95,0:00:57.94,Default,,0000,0000,0000,,7
Dialogue: 0,0:00:57.94,0:00:58.95,Default,,0000,0000,0000,,6
Dialogue: 0,0:00:58.95,0:00:59.92,Default,,0000,0000,0000,,5
Dialogue: 0,0:00:59.92,0:01:00.85,Default,,0000,0000,0000,,4
Dialogue: 0,0:01:00.85,0:01:01.96,Default,,0000,0000,0000,,3
Dialogue: 0,0:01:01.96,0:01:02.89,Default,,0000,0000,0000,,2
Dialogue: 0,0:01:02.89,0:01:03.100,Default,,0000,0000,0000,,1
Dialogue: 0,0:01:03.100,0:01:05.08,Default,,0000,0000,0000,,Give up?
Dialogue: 0,0:01:05.08,0:01:06.20,Default,,0000,0000,0000,,You should.
Dialogue: 0,0:01:06.20,0:01:07.51,Default,,0000,0000,0000,,It's not possible.
Dialogue: 0,0:01:07.51,0:01:12.64,Default,,0000,0000,0000,,But attempting to explain why\Nled famous mathematician Leonhard Euler
Dialogue: 0,0:01:12.64,0:01:15.100,Default,,0000,0000,0000,,to invent a new field of mathematics.
Dialogue: 0,0:01:15.100,0:01:18.65,Default,,0000,0000,0000,,Carl wrote to Euler for help\Nwith the problem.
Dialogue: 0,0:01:18.65,0:01:23.37,Default,,0000,0000,0000,,Euler first dismissed the question as\Nhaving nothing to do with math.
Dialogue: 0,0:01:23.37,0:01:25.14,Default,,0000,0000,0000,,But the more he wrestled with it,
Dialogue: 0,0:01:25.14,0:01:28.98,Default,,0000,0000,0000,,the more it seemed there might\Nbe something there after all.
Dialogue: 0,0:01:28.98,0:01:32.91,Default,,0000,0000,0000,,The answer he came up with\Nhad to do with a type of geometry
Dialogue: 0,0:01:32.91,0:01:38.26,Default,,0000,0000,0000,,that did not quite exist yet,\Nwhat he called the Geometry of Position,
Dialogue: 0,0:01:38.26,0:01:41.90,Default,,0000,0000,0000,,now known as Graph Theory.
Dialogue: 0,0:01:41.90,0:01:43.44,Default,,0000,0000,0000,,Euler's first insight
Dialogue: 0,0:01:43.44,0:01:48.51,Default,,0000,0000,0000,,was that the route taken between entering\Nan island or a riverbank and leaving it
Dialogue: 0,0:01:48.51,0:01:50.58,Default,,0000,0000,0000,,didn't actually matter.
Dialogue: 0,0:01:50.58,0:01:54.43,Default,,0000,0000,0000,,Thus, the map could be simplified with\Neach of the four landmasses
Dialogue: 0,0:01:54.43,0:01:56.63,Default,,0000,0000,0000,,represented as a single point,
Dialogue: 0,0:01:56.63,0:01:59.30,Default,,0000,0000,0000,,what we now call a node,
Dialogue: 0,0:01:59.30,0:02:04.20,Default,,0000,0000,0000,,with lines, or edges, between them\Nto represent the bridges.
Dialogue: 0,0:02:04.20,0:02:09.62,Default,,0000,0000,0000,,And this simplified graph allows us\Nto easily count the degrees of each node.
Dialogue: 0,0:02:09.62,0:02:13.22,Default,,0000,0000,0000,,That's the number of bridges \Neach land mass touches.
Dialogue: 0,0:02:13.22,0:02:14.60,Default,,0000,0000,0000,,Why do the degrees matter?
Dialogue: 0,0:02:14.60,0:02:16.83,Default,,0000,0000,0000,,Well, according to the rules \Nof the challenge,
Dialogue: 0,0:02:16.83,0:02:20.68,Default,,0000,0000,0000,,once travelers arrive onto a landmass\Nby one bridge,
Dialogue: 0,0:02:20.68,0:02:23.80,Default,,0000,0000,0000,,they would have to leave it \Nvia a different bridge.
Dialogue: 0,0:02:23.80,0:02:28.17,Default,,0000,0000,0000,,In other words, the bridges leading\Nto and from each node on any route
Dialogue: 0,0:02:28.17,0:02:30.59,Default,,0000,0000,0000,,must occur in distinct pairs,
Dialogue: 0,0:02:30.59,0:02:34.24,Default,,0000,0000,0000,,meaning that the number of bridges\Ntouching each landmass visited
Dialogue: 0,0:02:34.24,0:02:36.37,Default,,0000,0000,0000,,must be even.
Dialogue: 0,0:02:36.37,0:02:40.03,Default,,0000,0000,0000,,The only possible exceptions would be\Nthe locations of the beginning
Dialogue: 0,0:02:40.03,0:02:42.27,Default,,0000,0000,0000,,and end of the walk.
Dialogue: 0,0:02:42.27,0:02:47.22,Default,,0000,0000,0000,,Looking at the graph, it becomes apparent\Nthat all four nodes have an odd degree.
Dialogue: 0,0:02:47.22,0:02:49.19,Default,,0000,0000,0000,,So no matter which path is chosen,
Dialogue: 0,0:02:49.19,0:02:53.44,Default,,0000,0000,0000,,at some point,\Na bridge will have to be crossed twice.
Dialogue: 0,0:02:53.44,0:02:57.71,Default,,0000,0000,0000,,Euler used this proof to formulate\Na general theory
Dialogue: 0,0:02:57.71,0:03:01.72,Default,,0000,0000,0000,,that applies to all graphs with two\Nor more nodes.
Dialogue: 0,0:03:01.72,0:03:05.79,Default,,0000,0000,0000,,A Eulerian path \Nthat visits each edge only once
Dialogue: 0,0:03:05.79,0:03:09.16,Default,,0000,0000,0000,,is only possible in one of two scenarios.
Dialogue: 0,0:03:09.16,0:03:13.77,Default,,0000,0000,0000,,The first is when there are exactly\Ntwo nodes of odd degree,
Dialogue: 0,0:03:13.77,0:03:16.31,Default,,0000,0000,0000,,meaning all the rest are even.
Dialogue: 0,0:03:16.31,0:03:19.66,Default,,0000,0000,0000,,There, the starting point is one\Nof the odd nodes,
Dialogue: 0,0:03:19.66,0:03:21.77,Default,,0000,0000,0000,,and the end point is the other.
Dialogue: 0,0:03:21.77,0:03:26.09,Default,,0000,0000,0000,,The second is when all the nodes\Nare of even degree.
Dialogue: 0,0:03:26.09,0:03:31.23,Default,,0000,0000,0000,,Then, the Eulerian path will start\Nand stop in the same location,
Dialogue: 0,0:03:31.23,0:03:34.76,Default,,0000,0000,0000,,which also makes it something called\Na Eulerian circuit.
Dialogue: 0,0:03:34.76,0:03:38.46,Default,,0000,0000,0000,,So how might you create a Eulerian path\Nin Königsberg?
Dialogue: 0,0:03:38.46,0:03:39.30,Default,,0000,0000,0000,,It's simple.
Dialogue: 0,0:03:39.30,0:03:41.40,Default,,0000,0000,0000,,Just remove any one bridge.
Dialogue: 0,0:03:41.40,0:03:46.08,Default,,0000,0000,0000,,And it turns out, history created\Na Eulerian path of its own.
Dialogue: 0,0:03:46.08,0:03:50.20,Default,,0000,0000,0000,,During World War II, the Soviet Air Force\Ndestroyed two of the city's bridges,
Dialogue: 0,0:03:50.20,0:03:53.53,Default,,0000,0000,0000,,making a Eulerian path easily possible.
Dialogue: 0,0:03:53.53,0:03:57.29,Default,,0000,0000,0000,,Though, to be fair, that probably\Nwasn't their intention.
Dialogue: 0,0:03:57.29,0:04:00.78,Default,,0000,0000,0000,,These bombings pretty much wiped\NKönigsberg off the map,
Dialogue: 0,0:04:00.78,0:04:04.91,Default,,0000,0000,0000,,and it was later rebuilt \Nas the Russian city of Kaliningrad.
Dialogue: 0,0:04:04.91,0:04:09.08,Default,,0000,0000,0000,,So while Königsberg and her seven bridges\Nmay not be around anymore,
Dialogue: 0,0:04:09.08,0:04:13.36,Default,,0000,0000,0000,,they will be remembered throughout\Nhistory by the seemingly trivial riddle
Dialogue: 0,0:04:13.36,0:04:17.66,Default,,0000,0000,0000,,which led to the emergence of \Na whole new field of mathematics.