Problem #038 – bridges of Königsberg

You are on vacation and must find the most efficient way to cross all bridges. How will you do that?

Problem statement

If you like solving riddles and puzzles, it is likely that you have already encountered this puzzle. But even if you have, it is always good to go back and think about the classics. On top of that, I will formulate the problem in a slightly different way, so that you can be entertained for a bit even if you already know the more classic version.

Take a look at this satellite view from Kaliningrad, Russia, where I have highlighted seven bridges:

Your task is to figure out what route to take if what you want to do is cross all of the highlighted bridges at least once but, at the same time, keep the total number of crossed bridges as low as possible.

Having said that, what is the best route you can come up with?

(Just to be clear, I don't care about the length of the route – the number of miles/kilometres you would walk/drive – I only care about the number of bridges you cross.)

If you need any clarification whatsoever, feel free to ask in the comment section below.

In case you are wondering, the classic version of this puzzle is dubbed “the seven bridges of Königsberg” because that is what this place was called when a famous mathematician first dwelled on this problem.

Solvers

Congratulations to the ones that solved this problem correctly and, in particular, to the ones who sent me their correct solutions:

• Attila K., (Hungary)

Solution

There are seven distinct bridges that we want to traverse, so we know the shortest path has to go over seven bridges, minimum. What we will show is that, actually, we need to go over eight bridges in total in order to visit all seven bridges.

In order to show that is the case, consider the following figure:

In the figure above we can see that I numbered the four pieces of land to which the bridges are connected.

How many bridges does each piece of land connect to?

1. connects to 3 bridges;
2. connects to 3 bridges (as well);
3. connects to 3 bridges (as well); and
4. connects to 5 bridges.

Now we will use this information to show that it is impossible to create a path that visits all bridges exactly once.

Let us think about a hypothetical path we would do, in order to traverse all the bridges exactly once. More specifically, let us think about what happens in the middle of our walk. If we are in the middle of our path, when we enter some piece of land through a bridge, we have to leave that piece of land through another bridge. In other words, for each time we arrive at a piece of land, we also have to leave.

Suppose that the piece of land with the number $$1$$ is neither the departing point nor the arrival point. This means that whenever we reach land number $$1$$ we also have to leave it, which means that we need an even number of bridges connected to $$1$$, so that we are sure we can leave it whenever we arrive at it... But land number $$1$$ has an odd number of bridges, so land number $$1$$ must be the departing point or the arrival point.

However, all four pieces of land have an odd number of bridges, so all four pieces of land would have to be either the departing or the arrival point, which cannot happen because we can't depart from/arrive to multiple points. Therefore, a path that crosses each bridge exactly once won't cut it.

Now, can you come up with a path that crosses every bridge and only repeats one of them?

Here is an example of such a path:

This is actually one of the problems I solve with the participants in my workshop on recreational maths and I have to tell you: it is always fun to explore the mathematics behind such a misleading puzzle!

For those of you who are curious enough, what we touched upon on this problem is what we call an “Eulerian path”.