# 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.

Königsberg is german

"Thanks", commies!

I actually live in the middle of the map shown at the beginning

This is an amazing video

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.

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?????

Königsberg belongs to Germany

Huh, so by asking a seemingly pointless question he accidentally created a new field of mathematics. Cool

Swim

EULER’S METHOD EULER’S METHOD

Imposabro cus u can always be lft on Wong sid of berge

When you hyperfocus so hard help to invent a new field of maths by accident

Give konigsberg back to germany

The animation is super creepy

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!

A new field of mathematics when the reason you can’t do it is that it is an odd number

Couldn’t you swim or take a boat?

Kaliningrad technically

why can't any american pronounce ö? it's not a hard sound, we have the same sound in english.

Damn, you could've told me two bridges didn't exist anymore

Find the source of the river and walk around it.

At 1:55 it looks like Sonic

You could swim

Solved it, you just have to get a bit wet.

WOW! I never liked mayhematics but this is really great!

*Kaliningrad. Thank you.

The takeaway? "All problems can be solved with the appropriate and deliberate use of high explosives."

I solved this i just swam across in the water

Does it depend on where you start and finish?

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. 😁

So where are the bridge s now assuming the river still there and people need to cross

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

Hey 1 CE 2 AHAHHAHAHHAHA

It could be possible assuming that one could travel around the island at the right of the screen.

Seriously? Our class spent 1 and a half hours trying to solve this and now I know it's impossible smh

poor königsberg

problem solved bridges.

Just use a boat!

Isn't Kögnigsburg Kalingrad? Give it back to Germany 😂

Can’t solve a problem? Create a new mathematical field and boom there’s your answer ez

Königsberg für Preußen und das Vaterland!

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

Prussian Math

I know a way

Jump in the river

What city's are important in mathematics?

I solved the problem, all you gotta do is swim across the river

Solution:Construct one more bridge. Thank me later.3 and half hours wasted

Otto von Bismarck would like to know your location

K̶a̶l̶i̶n̶i̶n̶g̶r̶a̶d̶

Königsberg

always has been, always will be

Now to solve the Kjellberg Bridge problem…

It’s easy, just cross the river somewhere else, outside of the picture

It's possible. A girl named arushi solved it when she was nine

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

I’ve got a theory of how to do it

Graph theory!

You know it’s going to get good when you hear the name Euler.

What I learned,

If u get stuck on a math question blow up a bridge

damnnn!!! i figured it out!!! having top of the world feeling 😛 😛

Königsberg is rightful German clay!

The fact that Euler didn't consider this a math question REALLY highlights how much math has matured. Wow

My last name is Koenigsberg

This is a cool video, hope you make the Hamiltonian too. Thanks. :))

burn all mathematics books problem solve

Make a new bridge

I can do it, all you have to two is use a boat.

Wait,Euler? As in Euler's number?

1:56 it's a face

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.

these Renaissance dudes can't swim

not cool man

I see that you didn't call it "Kaliningrad". You are a man of culture

R.I.P. Königsberg/East Prussia

Blind and drunk Professor was lost on bridges on the home on night.

What??????

how was this animation made?!!!!!

Just heckin S W I M

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

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

Go around the river

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.

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.

What is königsberg? Are you talking about Krolowiec?

But still konigsberg is german

Wait so does going from the 2nd bridge to 1st to 5th to 6th to 3rd to 7th not work?

No one:

Ted: KONINGSBERG

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

This reminds me of ted ed riddles

Edgy…

Why don't you just call nodes dots, we Serbs do

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…..

I 💙 that person voice🗣….it relax my mind….thank u TED-Ed for those great videos

What if you go under one?

how about you just jump?

Wait… I was able to cross all bridges without double-crossing a bridge. Is this for real?

Agree, russians can't build (e.g. theories), they can only destroy.

And that’s how one-touch drawing games were made!

4:16 why does his mouth look like Ohio?

Solution? Build 1 more bridge…

Or you can just swim, didn’t have to invent a whole new field of mathematics but that’s good too

his nose is a legend