How the Königsberg bridge problem changed mathematics – Dan Van der Vieren


You’d have a hard time finding
Königsberg 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
in Königsberg? 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 wiped
Königsberg off the map, and it was later rebuilt
as the Russian city of Kaliningrad. So while Königsberg 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.

100 thoughts on “How the Königsberg bridge problem changed mathematics – Dan Van der Vieren

  1. I had a book about this Graph Theory and some exercises with solution, given by a kind student from a secondary school for gifted children. "I will finish this in 2 month" said I 3 years ago.
    Damn lazy life.

  2. Verdammt Königsberg!!!!!!!!!!!! Wegen dir musste ich das in der Schule lernen…Hätte mann nicht einfach eine große Brücke quer über eine der Inseln bauen können?????

  3. Just remember if you can't understand a concept in Math, obliterated the area of origins of the Map and ethnically cleanse Millions in a massive crime against Humanity but get let off the hook because the win the War!

  4. Königsberg may also have gained some sort of popularity, because of Immanuel Kant, the famous philosopher having lived there as well. At least that is the reason why I know it.
    Greatings from Germany. 😁

  5. i got this math quetion in year seven math and it told me to SOLVE it. So i clicked on this video

  6. if the last position isnt difined then u could stop any where right? that means that mathamaticians also need 'vocabulary' plus 'common sense' maaan!!

  7. Don't you Think Kaliningrad Bridge? (Now Russian City, It was prussian named Könisberg but it was built by Czech King "Přemysl Otakar 2.)

  8. Lol,destroy one bridge you crossed and make a new one. You didn't cross the new one yet! Well,just destroy the bridge and make a new one(in it's place) when there's only one more bridge left.

  9. But the problem isn't solved by removing any just one bridge? It has to be specific bridges, doesn't that make sense?

  10. This thing actually showed up in our computer book and spent the whole class trying to solve it XD

  11. You start from north-west, bridge 1, go to south, bridge 2, back from south (bridge 3) to north (bridge 4), pass the northern bridge of the 2nd island (bridge 5), cross from island 2 to island 1 on the 6th bridge. I can see a dock right there. A guy said you swim but suppose I cannot swim. So you go by boat south to the main land then cross the 7th bridge from south to the 2nd island…Or you can build an 8th bridge right in that corner where I said you take the boat (south east of 1st island) – after that it's just easy.

  12. If i may, you phrased the question wrong. i can cross all seven bridges once. but whats the outcome im not allowed to do? do i need to end up on the other side other side of the map?
    crossing all bridges once is possible. but what am i allowed and not allowed to do.

  13. We did this in 8th grade math class and I came up with both scenarios, but never thought to put them together…

  14. I swear all the classes combined in my whole 12 years of school life didn't teach me as much as these teded videos did…..

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