Isn't seven bridges problem trivial?

Question

What was the actual actual problem that led Euler to graph theory?

By looking even at non-simplified map like this

It is obvious that, if a landmass is connected by odd number of bridges, it cannot be only visited by crossing each bridge once - it must either be the start or the end of path. Here I see four landmasses with odd number of bridges, so it is obvious that there is no solution to the problem.

How could such a simple problem lead Euler to creating methods of analyzing such problem? And how could solution to such problem even get published?

My own guesses are that either mathematicians of the time were worried about some problems of rigour which I have missed or Euler just didn't imagine that the problem was unsolvable - instead he tried various unsuccesful paths and then simplified and reformulated the problem to get closer to the elusive solution.

Answer 1

The abstraction to land mases with an odd/even number of bridges is already a step that is not totally obvious - otherwise people without any prior knowledge of graph theory or this specific problem would not go ahead and try to sketch attempted solutions with trial and error (and they do).

Answer 2

If you consider this problem trivial today, this is in part due to the fact that Euler created the language of graph theory that makes it easy to formulate this kinds of reasoning. Indeed the general fact that a connected graph admits an Eulerian path (if and) only if at most two vertices have odd degree is a very simple theorem of graph theory. But in Euler's time, mathematics simply did not have much language to discuss or solve this kinds of problems, so it is to the credit of Euler that he put in place the abstractions necessary to do these things cleanly. This was a more fundamental step than developing more involved theorems once the language is in place.

You may object that you did not need the language of graph theory to formulate an obstruction showing that the problem is unsolvable. Well, you may not have used the same language, but the argument is based exactly on the same abstractions that lead to graph theory. The very fact of formulating the problem in terms of landmasses (vertices) linked by bridges (edges), and reducing the route followed to a sequence of bridges crossed, is precisely the abstraction that makes the problem tractable, and indeed quite easy. You need not know about graph theory, or even be aware that you are doing this abstraction when formulating the obstruction, but if you try to analyse carefully what makes your argument work, you will find that in the end this abstraction is the crucial factor.

It seems unlikely many people cared about the seven bridges problem at all before Euler dealt with it. Or maybe some did, but after a bit of reflection concluded that there is (probably) no solution, and stopped worrying about it; history would certainly have forgotten them. The great insight of Euler is that a powerful abstraction is involved that can be applied very effectively to a wide range of more difficult problems.

Of course if Euler hadn't invented graph theory someone else would have; it is a fairly natural thing to do. But it happened to be Euler who did it first.