
Seven Bridges of Königsberg - EndXA
https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg
======
archgoon
The bridges today in modern Kalningrad.

[https://www.google.com/maps/place/Knaypkhof,+Ulitsa+Kanta,+K...](https://www.google.com/maps/place/Knaypkhof,+Ulitsa+Kanta,+Kaliningrad,+Kaliningradskaya+oblast',+Russia,+236039/@54.7046814,20.5115908,16z/data=!4m13!1m7!3m6!1s0x46e33d8d4b7c21a9:0x5050960016126ed3!2sKaliningrad,+Kaliningrad+Oblast,+Russia!3b1!8m2!3d54.7104264!4d20.4522144!3m4!1s0x46e31611283b3c4d:0x7112c08b68959f4a!8m2!3d54.7069392!4d20.5086368)

It appears that the geometry has changed since Euler's time, or the diagram is
incorrect. It appears that is the larger ( Lomse/ Oktyabrsky ) island that has
the 4 bridges from the mainland rather than the smaller one (Kneiphof).
Ironically; the topology appears preserved, so it doesn't actually matter,
which underscores the point of Euler's achievement.

~~~
dmix
The cathedral on the island is beautiful and contains Immanuel Kant's tomb. It
was bombed pretty severely in WW2, before that there were a bunch of buildings
on the island:

> After the war, the cathedral remained a burnt-out shell and Kneiphof was
> made into a park with no other buildings. Before the war, Kneiphof had many
> buildings. One of the buildings was the first Albertina University building,
> where Immanuel Kant taught, which was situated next to the east side of the
> cathedral.

[https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Cathedral](https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Cathedral)

Picture of it prior to WW2 (featuring more bridges?):
[https://upload.wikimedia.org/wikipedia/commons/3/38/Modell_D...](https://upload.wikimedia.org/wikipedia/commons/3/38/Modell_Dominsel_K%C3%B6nigsberg.jpg)

~~~
trhway
The cathedral got lucky as its skeleton, the walls, whats left after WWII
fighting and bombing, were allowed to survive through the Soviet times. During
198x were was a flea/farmers market inside and behind it.

The Castle didn't get that luck -
[https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Castle](https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Castle).
It got bulldozed, and the boondoggle of Soviet glory - the "skyscrapper" (21
story in Kaliningrad region in 198x was almost mindblowing, add to that that
it is on the top of the Castle hill, so it looms large over the city) - is
still unfinished and

([https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Castle#Current...](https://en.wikipedia.org/wiki/K%C3%B6nigsberg_Castle#Current_situation)
) "Continuation of development was stopped in the 1980s as the massive
building gradually sank into the structurally unsound soil stemming from the
collapse of tunnels in the old castle's subterranean levels. Many people call
this the "Revenge of the Prussians" or "The Monster"."

These underground tunnels (Kenigsberg and Prussia as whole has a lot of them)
were back then and i'd guess even more today the scene of active exploration,
in particular for hidden Nazi stolen treasures (e.g. "Amber room"),
weapons/munitions/etc. and just out of sheer curiosity ( after all it is
centuries of history starting from Teutonic Order). So you sometimes would
find yourself facing a problem of "N tunnels in three dimensions" with some of
them flooded/leading into unknown/etc.

------
myleshenderson
My grandfather gave me a puzzle when I was 8 or 9 years old where the goal was
to trace a path through each of the "walls" in the diagram below without the
path crossing itself:

    
    
      ---------------------
      |        |          |
      ---------------------
      |    |        |     |
      ---------------------
    

I played with it off and on for years and suspected that it was impossible. I
realized that it is in fact impossible when I took discrete mathematics in
college and we covered the Seven Bridges.

I've passed this on to my kids but only let them play with it for hours until
revealing that it's not possible and getting into the theory of it.

I don't think my grandfather knows this is impossible, and I haven't yet
remembered to tell him.

~~~
Stratoscope
In fact, this five room puzzle is mentioned in the Königsberg article and also
has an article of its own:

[https://en.wikipedia.org/wiki/Five_room_puzzle](https://en.wikipedia.org/wiki/Five_room_puzzle)

Your grandfather would likely enjoy discussing it with you!

~~~
cjauvin
Wow thank you! So I just learned the proper name of this "puzzle", which my
father also showed me a long time ago (and for which I used to fill pages and
pages of trials even though I knew that it had no solution), and which I tried
to render as an "applet toy", a couple of years ago on my blog (see my other
comment below).

~~~
TuringTest
It's interesting that there exists a solution (shown in the Wikipedia
article), if you build the rooms on a torus. That would be the equivalent of
digging a tunnel between two rooms.

------
tom_mellior
The discussion of the "Variations" section is interesting. Apparently the
person who inserted it (and had it removed several times) won't give a
citation but also won't come out and say that they made the examples up
themselves. Nobody ever calls out the useless introduction of German terms in
the examples, and more importantly that (a) "Ritcher" is not a German word,
and (b) even if it were a misspelling of some German word, it would not mean
"church".

~~~
justin66
Having knowledge of the material and writing it down is considered a negative
when it comes to creating Wikipedia content.

~~~
tom_mellior
The way the section was phrased (it's removed now) it appeared to refer to
some external source. The sentence "It is understood that the problems to
follow should be taken in order, and begin with a statement of the original
problem:" would make no sense otherwise. But if there is such an external
source, it should be cited. This is the main problem. It's not about Wikipedia
being a boring stickler for rules.

------
cjauvin
I was fascinated by a "puzzle version" of this problem as a child, and a
couple of years ago I wrote some little Processing applets to explore the idea
of "gamifying" it:

[https://cjauvin.blogspot.com/2012/02/implicit-
bridges.html](https://cjauvin.blogspot.com/2012/02/implicit-bridges.html)

[https://cjauvin.blogspot.com/2012/02/eulerian-
hoax.html](https://cjauvin.blogspot.com/2012/02/eulerian-hoax.html)

------
throwaway287391
It's interesting that it's apparently the problem that laid the foundation for
modern graph theory, but the problem can't actually be specified as a graph
under its usual modern definition... can it? At least not without some
shenanigans that would make it more complicated than "node = landmass, edge =
bridge", as it's depicted in the standard diagram shown on Wikipedia.

A graph is usually defined as a set of nodes and a set of edges, where an edge
is a pair of nodes (an ordered pair if it's a directed graph). The "set" of
edges (bridges) of the "graph" in this case contains duplicates -- there are
two edges with the same pair of vertices, two bridges crossing between the
same pairs of landmasses. And the two pairs of duplicated bridges are
essential to the problem setup. So AFAICT it can't be fully specified as a set
of edges; it would have to be a multiset.

(I've seen this problem before but only just noticed this when reading
Wikipedia now -- let me know if I'm missing something.)

~~~
ColinWright
A "graph" can sometimes implicitly mean a "Simple Graph" which has neither
loops nor multiple edges between pairs of vertices, but in some contexts can
mean "Multi-Graph", which does allow multiple edges and loops. So the usual
conversion of the bridges problem to a network gives us a "Multi-Graph" ...
the distinction is often left to be inferred, because it's usually obvious (to
the skilled practitioner).

One way around this to end up with a simple graph is to include a vertex in
the middle of each bridge. Then each edge has one end in the landmass and the
other end in the middle of the bridge. That equivalence/conversion shows that
the distinction, in this case, is effectively unimportant.

------
darrenf
Some friends and I took a holiday to NYC a few years ago, calling it "Project
Königsberg" as we attempted to travel all public transport entry and exits to
Manhattan without duplication, over the course of 3 days. I was glad not to be
involved in its planning, but very happy to participate!

------
RhysU
I loved the 2015 SIGBOVIK paper applying this idea to Pittsburgh bridges. Page
21 of
[https://archive.org/details/SIGBOVIK2015/mode/2up](https://archive.org/details/SIGBOVIK2015/mode/2up)

------
bmc7505
William Hamilton has an interesting variation of Euler's Seven Bridges puzzle
[0] which asks whether there is a path visiting every vertex of a polyhedron
exactly once. Strangely, while the complexity of detecting Eulerian paths is
linear in the number of edges [1], the complexity of detecting Hamiltonian
paths is NP-complete [2]!

Hamilton (who was also fond of crossing bridges [3]) developed some
interesting algebraic structures to study the polyhedron problem [4]. I
recently became interested in Eulerian and Hamiltonian paths after taking a
class in Graph Representation Learning [5]. In it, I learned there are many
interesting connections between algebra and graph theory [6] and started
writing a library called Kaliningraph to study some of those connections [7].

[0]:
[http://eulerarchive.maa.org/docs/originals/E053.pdf](http://eulerarchive.maa.org/docs/originals/E053.pdf)

[1]:
[https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer's_alg...](https://en.wikipedia.org/wiki/Eulerian_path#Hierholzer's_algorithm)

[2]:
[https://en.wikipedia.org/wiki/Hamiltonian_path](https://en.wikipedia.org/wiki/Hamiltonian_path)

[3]:
[https://en.wikipedia.org/wiki/Broom_Bridge](https://en.wikipedia.org/wiki/Broom_Bridge)

[4]:
[http://www.kurims.kyoto-u.ac.jp/EMIS/classics/Hamilton/PRIAI...](http://www.kurims.kyoto-u.ac.jp/EMIS/classics/Hamilton/PRIAIcos.pdf)

[5]: [https://cs.mcgill.ca/~wlh/comp766/](https://cs.mcgill.ca/~wlh/comp766/)

[6]:
[https://www.cs.yale.edu/homes/spielman/sagt/sagt.pdf](https://www.cs.yale.edu/homes/spielman/sagt/sagt.pdf)

[7]:
[https://github.com/breandan/kaliningraph](https://github.com/breandan/kaliningraph)

------
JorgeGT
Soviet solution: bomb two of the bridges, therefore allowing an Eulerian path.

~~~
gherkinnn
You mean an Alexandrian solution (to a Gordian knot?)

~~~
lisper
I think the GP was intended to be funny. (And, judging by the downvotes, HN
still has zero tolerance for humor.)

~~~
pfarrell
Well, the HN/engineer solution would be to propose the building of a new
bridge.

Then we could have a big discussion about funding strategies, metallurgical
esoterica, cutting edge topological algorithms for determining best place to
add a vertex in the graph, and/or history of Roman land jurisdiction. And we’d
all learn something.

~~~
usrusr
The problem states seven bridges, not seven or more. Clearly, the engineering
would be a tunnel.

(the beaver solution would be a dam. "List of mathematical problems solvable
by beavers")

------
hprotagonist
i was extremely lucky and had a great math teacher who pulled this one on us
in 10th grade.

Good times.

------
HugoDaniel
Who cares about bridges when you can swim ?

