Talk:Seven Bridges of Königsberg

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Date of Euler's publication: 1735 vs. 1736[edit]

I was wondering about the date given in this article (1735). In the German wikipedia, the year 1736 is given. A simple google search gives almost twice as many hits for 1736 compared to 1735. However, I cannot investigate this any further. Does anyone bother to check this or do you want to keep 1735? — Preceding unsigned comment added by (talk) 12:30, 7 June 2013 (UTC)

The extra problems are not consistent[edit]

The "Bridge 8" part speaks about Blue prince wanting to go from his castle to the guesthouse, "Bridge 9" says the Red prince wants to stop the Blue prince from going doing a circuit. (talk) 20:20, 27 December 2012 (UTC)

Fixed that. (talk) 11:53, 23 April 2013 (UTC)

I think the solutions to the extra problems contains a mistake[edit]

The first problem (8) solution's bridge seems to be in the wrong place. It should be where the tenth bridge is put. If the bridge is where it is shown to be, then the bridge between the orange and white node is not crossed, or if it is, then the orange node is the finishing place. This is clear because the white node is the final resting place so should have an odd number of bridges, while the orange crossover node needs to have an even number, one in, one out. The explanation states that it is required that there be either zero or two nodes of an odd degree. Both the given explanation and my alternative fulfil this criterion. But it seems that both the starting and finishing nodes must be the odd ones. Does someone want to say that I am wrong, or should it be changed? Tsop 10:26, 12 October 2007 (UTC)

I think its OK, the orange node the Gasthaus is the desired final resting place, not the white node. --Salix alba (talk) 11:09, 12 October 2007 (UTC)

Looked at and Mark Foskey

The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point.

I just got a math test back that asked about the Bridges of Konigsberg and got the question wrong because i stated the problem like this. According to my professor, the question does not require that you return to the starting point. i haven't made a trip to the library yet, but the few websites i've checked out seem to pose the question without the "return to the starting point" qualifier. is this an error? (adam)

   Adam -- Was the question about a circuit 
   or a tour?  The "tour" version of the problem
   does not require return to the starting point;
   the "circuit" version does. 

Euler showed that a path of the desired form is possible if and only if there are exactly two nodes (dots in the picture above) that have an odd number of edges touching them. Since the graph corresponding to Königsberg has four such nodes, the path is impossible.

What if there are no such nodes (for example, a triangle of three nodes, three edges)? Shouldn't it be "iff there are either 2 or 0 nodes ..."? Chas zzz brown 07:41 Mar 15, 2003 (UTC)

yup, it should. -- Tarquin 09:43 Mar 15, 2003 (UTC)

But there is a solution. Your model is simply too small. Try to look for the big picture!

Category removal[edit]

I think this shouldn't be in the category of "Localities with numbers in their names". There is no locality named "Seven Bridges of Königsberg". There is only a locality named Königsberg, and it has seven bridges in it. The bridges by themselves are famous because of this problem, but don't constitute a locality.

Discuss. JIP | Talk 10:37, 9 Mar 2005 (UTC)

Missing Image[edit]


Where is the new image? (Image:7_bridgesID.png) — Xiong (talk) 02:32, 2005 Mar 20 (UTC)


Oh, this is beyond bizarre. Apparently, a 288px thumb is not allowed. 287px is okay; so is 289px.Also, 144px and 500px sizes are okay. WTF?? I swear I saw the image (at 288px) when first I inserted it. — Xiong (talk) 10:07, 2005 Mar 20 (UTC)

Never mind. — Xiong (talk) 02:59, 2005 Mar 21 (UTC)


My mother had a small collection of mathematical puzzle-books, and I believe I saw this variation in one of them. I have not seen any of those books for several years, and may not have read the telling of this variation in 30. The diagram and words are my own, though not the idea. I wish I could uncover the source; indeed, upon her death, these books were practically my only inheritance. But they are now unavailable to me, being several thousand miles away. — Xiong (talk) 10:17, 2005 Mar 20 (UTC)

Textbook style[edit]

This article, at the moment, seems to me to be more suited to a text book than an encyclopaedia. Specifically, there are problems given (which I have no problem with in this context, since the article is essentially /about/ those problems), but I don't think the answers should be given separately, but rather worked into the explanation of the problems. Also, the tone of the 'solutions' article is exteremely textbooky (it speaks directly to the reader, whereas an encyclopaedia generally uses the neutral voice). Perhaps it should be re-worded, and/or some of the extra problems moved to a Wikibook on Graph Theory? LVC 05:56, 15 October 2005 (UTC)

I agree. I ran into this problem with my own edits just now. I started out wanting to make just one substantial change (referring to graph theory as a subfield of combinatorics, rather than topology), but in the process of doing so ended up changing some of the wording. It was hard not to write like a textbook! I hope the revised Sections 1 and 2 are (slightly) more encylopedia-like in tone. That said, I don't know how to integrate problem and solution any more than what is there right now.JLeander 23:16, 20 January 2006 (UTC)

An encyclopedia is supposed help people learn or understand about a certain topic, right? So if it's written in a way that is not directed towards the person, it will most likely be harder for that person to understand it. When I read this article I immediatly "got it" because it was written in a way that felt like they were talking to me. Some of the more "encyclopedic" articles that I've read were harder for me to understand because they were using the "neutral voice". The best articles are the easiest to understand. Rsercher (talk) 17:16, 15 June 2011 (UTC)

Phrasing issues[edit]

I think that the wording in the most recent version (modified by is less clear than the previous one. In particular, I think the phrase "and is thus the complete opposite of what is required" is not mathematically precise, and misses the point, which is that the existence of any node of odd degree forbids the existence of an Eulerian circuit. Accordingly, I am going to revert to the version modified by Icairns, 09:24, 24 January 2006. I hope this is OK; I am a Wikipedia novice and don't mean to step on any toes.JLeander 05:20, 28 January 2006 (UTC)

State of the bridges now?[edit]

Someone told me once that after the Russians took over the city in 1945, they built a bridge (the joke was that it was to spite Euler) which now makes an Eulerian circuit possible. Can anyone confirm this and/or add details? --Saforrest 02:54, 30 May 2006 (UTC)

My guess is that by the time Russians took over the city in 1945 all the bridges had been destroyed. So they built (restored/rebuilt/built anew) not just a single bridge but dozens. According to the corresponding Russian page three of the original bridges still exist. Two bridges were not restored after WWII and a new bridge replaced two other bridges in 1970's. Plus some other bridges have been built nearby (outside the area delimited by the original seven bridges but within the old "Eulerian" borders of Königsberg). Also, the border of the city has changed since Euler's times. So the question of how many bridges are there in Königsberg (Kaliningrad) now is not a simple one. -- 20:05, 4 July 2006 (UTC)

Here is a link to Google Satellite view of the area today:,20.5163074,2217m/data=!3m1!1e3 SlowJog (talk) 03:21, 22 May 2015 (UTC)
I don't know what the actual state was in 2000, but the article says there were 5 bridges, and the map in that section shows 8. Also, if today's map were to be zoomed out a little more than the one in that section, there would be 9. It isn't clear from the map where the city boundaries are, so it could be that 5 of the bridges are within the city limits. In either case, that section, the image, or both need updating. SlowJog (talk) 03:48, 22 May 2015 (UTC)

Variations, Present state sections[edit]

The following section is entirely inappropriate, because it's written like a children's textbook or puzzle book instead of an encyclopedia article, and more importantly it has no sources so it's effectively original research. The hiding of the solutions also goes against the spirit of WP:NDT. —Keenan Pepper 18:47, 11 November 2006 (UTC)

No, your deletion is inappropriate. This section has been in the article a long time and adds value to the problem. If you have concerns, edit the section to address them. I don't endorse wholesale deletions on skinny justification. Fix this up to suit yourself and put it back. Okay?
I don't question your addition of the "Present state" of the bridges, even though this is totally irrelevant to the classic problem. Play ball. John Reid ° 20:25, 11 November 2006 (UTC)

I have no idea where these variations come from, so I have no sources, so I can't write anything verifiable. What do you suggest I do? —Keenan Pepper 20:32, 11 November 2006 (UTC)

There is nothing to verify here. This is an illustration of a problem in graph theory. Are you unable to follow the reasoning? This is like illustrating the topic of Arithmetic with the sum 7 + 11 = 18. It's just a set of facts, that's all. Go ask a combinatorialist -- one you personally trust -- if you disagree. I'm sure you can find one at your local university. Okay?
Meanwhile, you can think how to connect your insertion of bombed and rebuilt bridges with the topic -- which has almost nothing to do with the real city and everything to do with graph theory. You didn't even include a diagram. John Reid ° 06:30, 12 November 2006 (UTC)
It seems like you're trying to score points off me rather than improve the article, but I'll reply anyway. Of course I can understand the problems, and they seem like good problems for teaching graph theory to children, but that's not the point. My point is that including them violates core Wikipedia policies like WP:CITE and WP:NOR. You compare it to using an example of an arithmetic problem for the article on arithmetic, but that comparison is invalid because in this case the topic is not a general concept that could benefit from examples. This article is about one specific, notable problem, and these are related but different problems, with no claim of notability.
I think it's absurd to say that a section on the current condition of the seven bridges of Königsberg is inappropriate in an article titled "Seven Bridges of Königsberg". I didn't include a diagram because I couldn't find one with a free license, and my skills as a graphic designer are poor to nonexistant. Anyone is of course welcome to provide one. —Keenan Pepper 05:59, 14 November 2006 (UTC)
The current condition of which you speak is the current condition of the town. The current condition of the problem is exactly what it has always been since Euler formalized it. This is an article about the problem. There is an article on the town of Kaliningrad; you're welcome to talk about the real bridges there. The variations shown here have long illuminated and extended this topic. They were here before I joined. They are not notable in themselves; they merely explain the notable problem. They may seem childlike to you but I find them appropriate. You are welcome to edit them to raise whatever level of maturity you perceive.
I will tolerate your bridges if you tolerate these. It's just that simple. For that matter, if you scribble your bridges in any legible way, I will do my best to diagram them nicely. John Reid ° 09:56, 15 November 2006 (UTC)
Why do you call them "[my] bridges"? They have nothing to do with me personally. The web pages I cited have maps.
You haven't responded to my accusation that the variations violate WP:CITE and WP:NOR. Are you arguing that they satisfy those requirements, or that the requirements don't apply in this case? The problem isn't that they're childlike, the problem is that they were just made up by some anonymous person and Wikipedia isn't the place for them. —Keenan Pepper 20:05, 15 November 2006 (UTC)

I agree that the section on variation adds to the article. I also agree that the style is inappropriate. This is not a quiz-book. I suggest that the references to various Princes thoughts and wants and the rubbing of various bodily parts are removed, and that the solutions are made non-hidden. The information in the section is useful. The current presentation just distracts from that. --Regebro 23:35, 15 November 2006 (UTC)

You do not have to cite every word in every article, Keenan. You don't even have to cite a long-winded exposition. You have to be able to support every claim made. It's perfectly acceptable to create original content; just not to advance any original thesis. If you don't say that these variations claim some innovation in graph theory, you need to pack up that citation wagon and move it along.
Keenan, your sources disagree about the present state of the bridges in the real city of Kaliningrad. This makes both sources suspect. One reference says, interestingly, the cathedral was on the central island, not the pub. You might need to change a lot of things to make the variations agree with this little historical tidbit. Every classic bridge was named, too, but this would really screw up the article, since the problem has nothing to do with colored edges.
You are factually incorrect when you say the variations were "made up by some anon". They were heavily edited by an anon (and many others) but if you actually go back through history, you see they were put in there by the redoubtable Xiong. I've come across this fellow before; he's done interesting things in templatespace. He seems to have had a talent for pissing people off and, as usually happens, got pissed off himself and left the building. I don't see that as any reason to torch his work -- especially since it's not his any longer.
They are "your" bridges because you put them there; Euler did not. They have no obvious place in an article on graph theory. Either you understand this or, I suspect, you are not a mathematician. Can you not understand that if Kaliningrad went through an orgy of bridge-building or demolition; if the whole city were destroyed in nuclear holocaust; Köenigsberg would, has, and always will have exactly seven bridges? The only place where bridges can be added or subtracted is in our minds.
Regebro, the language is colorful. I like colorful. I don't like long, gray pages with nothing to look at, nothing to stimulate the imagination or make me feel life is potentially worth living. I agree that the solutions don't need to be stuffed into show/hide boxes; I abominate them. It's enough to state the variations, stick in a standard spoiler warning, and copy in the stuff from /key.
Your talk about rubbing "body parts" is misleading; there's none of that there. The talk about why the princes and the priest do as they do motivates the variations. Motivation is an important part of decent writing; if you cannot show why there is no point showing how. I agree that the language is overly florid, even rankly overgrown. Prune it. I wouldn't know where to begin without destroying, as the priest fears, the article's Weltanschauung.
I suspect that in order to satisfy both of our concerns, you will have to rewrite the variations so as to have engineers building additional bridges, rather than princes; and bombers destroying them. I can't say I think it will improve anything. Anyway, if anybody proposes a rewrite that requires additional maps, please scribble out graphs and I will work them up.
Meanwhile, I'm restoring the variations. Anybody can edit but nobody can destroy without cause. We've discussed this enough. John Reid ° 01:17, 16 November 2006 (UTC)

In the variants problem it is also a correct solution to draw the bridge from red to orange forcing the start and end points of the walk to be blue and white.


Removed this copy of article text; see history for old versions. John Reid ° 20:38, 16 November 2006 (UTC)

Third opinion[edit]

I found my way here from Wikipedia:Third opinion.

  • First, I have to say that I don't know that javascript is appropriate in articles at all. It doesn't work in all browsers, presents difficulty for the sight-impaired, and many folks disable it. Isn't there a policy on such things?
  • I think the Variations section above is pretty cool, well-written, and amusing, but I must admit it fairly screams "original research"!! If it were re-written, perhaps it would be acceptable.
  • Finally, are variations of the problem even notable enough to describe in detail? Reading the article, it doesn't seem like that to me. Maybe this section should be replaced by a short paragraph summarizing the variations, with a link to the solutions as above. That would allow the content to be kept in a fashion, without actually having it so contentiously in the article. -Amatulic 01:03, 16 November 2006 (UTC)
Please explain, in short words, what kind of original research or novel thesis is advanced by the variations. Also, please tell us -- and I assure you, I don't mean to be rude -- if you have any particularly strong math background. I ask because mathematicians are accustomed to hypotheticals and puzzles. When you were in school, did you like word problems?
You suggest a rewrite; why don't you attempt one? I never object to editing in general, only to wholesale deletion. Note that I did a bit of rewrite myself; I don't like those show/hide boxes anywhere, never have. Notability is not an issue here; the variations are not the subject of the article, they only illuminate it.
In case you're not too familiar with graph theory, let me present a simpler analogy. Reasoning by analogy is always strained but it's simple. Please don't try to argue it back and forth. I only intend this as a gateway to understanding the variations.
If the circular Kingdom of Nod is 7 km across then its border is about 22 km long.
Let us say I wish to illustrate, with a picture, Pi. My references tell me that this is the ratio of a circle's circumference to its diameter. I decide to produce a graphic, so: Image:Nor-demo-1.png. This illustration of Pi is ugly, uninformative, and unmotivated but it is not original research. Not even when the caption is included have we crossed that Rubicon.
Let me set up the straw man:
That graphic is original research. There is no Kingdom of Nod. Also, I see no citation that establishes the circumference of a 7 km circle colored red with a little checkerboard pattern.
This is, of course, an absurd criticism. Of course there is no Nod; it's purely hypothetical. While one might be able to find, by digging, some peer-reviewed journal paper that states π applies to a 7 km circle, it might be difficult; let us say such does not exist. Nonetheless, it is a straightforward calculation. We -- we, here -- are permitted to make this. The stroke and fill of the circle are totally irrelevant; the straw man is simply talking through his hat.
Xiong's illustration -- not the abstract diagrams but the map-ish graphic -- is rather cute -- too cute for my taste but not so much I can't bear it. His language, which seems to have survived untouched through history, is droll. To a point I support this; I like colorful writing. To another, it is just too cute. For at least the fourth time in this thread, I'll be happy to see a rewrite. I will bet a dollar to a doughnut that the cuteness is the main motivator of criticism here. I've seen worse in this project. You need to distinguish form from content.
I've asked before, I'll ask again: Please go to a person you consider to be a competent graph theorist and ask him if the variations constitute any sort of original research -- any sort at all. Ask him if some claim is being made, some innovation in the field. It looks like a straightforward hypothetical exposition to me -- a word problem. John Reid ° 20:33, 16 November 2006 (UTC)

As a professional combinatorialist, I agree with John Reid that the "Variations" section does not constitute original research. On the other hand, I don't think it's of much significance. The importance of the Seven Bridges problem is that it is a problem that originated in real life and led in part, thanks to Euler, to the establishment of a distinct branch of mathematics. It's easy to come up with lots of variations on the problem, but I think they are more suitable for a homework assignment in a graph theory course than an encyclopedia article. So I think that this section should probably be omitted, although I don't feel particularly strongly about it. JLeander 14:00, 17 November 2006 (UTC)

Thank you, Jeremy. If we've disposed of the NOR objection then all we're left with is a matter of taste. I feel the problem is complete in itself but the article then leaves one unfulfilled; it is an explanation of exactly one special case. When I read through the variations -- or even think about any kind of variations on the problem, bridges built, demolished, or whatever -- this helps me to understand what Euler was driving at.
I agree the variation maps are too cute. The language in which the variations is stated is florid. I will be happy to fix the former if someone will improve the latter.
Also, my offer stands to illustrate the "current state", if somebody will express that as a problem. In fact, to show good faith, I'm creating Talk:Seven Bridges of Königsberg/Today. Let's see what we can put together. John Reid ° 03:09, 18 November 2006 (UTC)
The "variations" section is not "original research" in the sense that it is groundbreaking new research or that it is of questionable correctness. It is "original research" and "non-notable" in the Wikipedia sense that no reliable source attests to its significance or relationship to the original topic. As another writer wrote above, it might be a fine homework problem or whatever, and it illustrates some points in graph theory, but we have no reliable source linking it to the Seven Bridges of Koenigsberg. WP policy demands that we remove it. --Macrakis (talk) 22:49, 1 February 2011 (UTC)

A Solution:[edit]

Q: It is possible to walk with a route that crosses each bridge exactly once, and return to the starting point?

A: Yes, if you do it in winter when the rivers are frozen over. (LOL, Am I right or what?)

~Mix. —The preceding unsigned comment was added by Mix Bouda-Lycaon (talkcontribs) 09:47, 25 January 2007 (UTC).

Puzzle Name[edit]

What is general name for puzles in which you have to redraw a figure without picking your pensil off the paper, retrace a line, or folding a piece of the paper?--Anthonynow12 17:20, 4 February 2007 (UTC)

A topology problem. I dunno if there's a nick-name for it. I Haven't seen many. Though i know how to connect a 3x3 grid with 4 connected lines. —The preceding unsigned comment was added by Mix Bouda-Lycaon (talkcontribs) 06:41, 6 February 2007 (UTC).

node mistake[edit]

the puzzle is possible if the nodes have 0, 2 OR 3 odd nodes, the article only says 0 or 2, i will fix this —Preceding unsigned comment added by (talk) 11:36, 21 August 2008 (UTC)

Only 0 or 2. Either you start and end at different vertex (then the endpoints are of odd degree and every vertex is even), or you have a circuit (and then every vertex including the starting point is of even degree). WLior (talk) 20:54, 22 August 2008 (UTC)
It is impossible for a network to have 3 odd nodes, or any other odd number for that matter. A simple proof of this is that an odd number of odd nodes would give an odd total number of endpoints, but every segment must have exactly 2 endpoints, so the total must be even. (talk) 21:51, 27 December 2010 (UTC)

Confusing Text[edit]

Two of the seven original bridges were destroyed by bombs during World War II. Two others were later demolished and replaced by a modern highway. The three other bridges remain, although only two of them are from Euler's time (one was rebuilt in 1935).[2] Thus, there are now five bridges in Königsberg (modern name Kaliningrad).

In terms of graph theory, two of the nodes now have degree 2, and the other two have degree 3.

Two demolished, two replaced by one, and the remaining three, makes four. The next line says "five", and the next paragraph mentions only four again.--WPaulB (talk) 17:48, 28 February 2010 (UTC)

(1) If two bridges are replaced by one highway, the highway still has to cross the river in both places so there are still two bridges (which plus the three remaining ones makes five as stated). (2) The "nodes" in the problem are the landmasses, not the bridges. (talk) 21:54, 27 December 2010 (UTC)
In an article about graph theory, telling me that two edges between a pair of nodes can't be replaced by one edge is absurd. It is entirely possible to remove two bridges and replace it with one wider bridge that holds a highway. That's the essence of flow diagrams, no - weighted edges? Second, highways have a beginning and end. It is entirely possible that a highway can start or end on one landmass, and not cross a river twice. if you want to make this unambiguous, display it on a map. --WPaulB (talk) 13:46, 7 February 2012 (UTC)

File:Konigsberg bridges presentstatus.png Nominated for Deletion[edit]

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Important: the walk need not start and end in the same spot.[edit]

Otherwise it is simple to solve the "problem". — Preceding unsigned comment added by (talk) 22:08, 1 February 2012 (UTC)

I take that back: in the picture as shown, all islands have odd degree. So we should emphasize that the walk need not start and end in the same spot. That is what makes these particular bridges special.


It is (as of 26 February 2006) possible to walk the (six or seven) bridges, starting from the northern shore and ending on the island in the middle (the Kneiphof). [1] Double sharp (talk) 07:40, 18 March 2012 (UTC)

The exposition of the problem is not precise enought[edit]

"The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges."

We should precise that the exposition of the problem is the map, because if the seven bridges were placed differently, we can find solutions has above :

    /               |              \
----                D               ------------
    \               |              /