Why is the Seven Bridges problem unsolvable, and how to answer the Seven Bridges Problem?

In the seven-bridge problem, there are odd curves intersecting at four intersections, so the problem is unsolvable.

Because he has 4 odd points, in other words, if you want to draw a stroke, then for each node the number of entries should be equal to the number of exits (except for the beginning and end points), and the Seven Bridges problem does not satisfy this.

Because it has more than two singularities.

Euler used dots to represent islands and land, and the lines between the two points to represent the bridges that connected them, reducing rivers, islands, and bridges to a network, and reducing the problem of seven bridges to a problem of judging whether a connected network could be drawn in one stroke. Not only did he solve this problem, but he also gave that the sufficient and necessary conditions for a single stroke to be connected to the network are that they are connected to each other, and the number of vertices (the number of arcs through this point is an odd number) is 0 or 2

Encyclopedia (Seven Bridges Problem)...Very detailed.

One of the famous mathematical problems of the 18th century. In one of Königsberg's parks, seven bridges connect the two islands of the Pregel River with the banks of the river (pictured). Q: Is it possible to start from any of these four landmasses, pass through each bridge exactly once, and then return to the starting point?

Euler studied and solved this problem in 1736, and he reduced the problem to the "one-stroke" problem shown in the picture on the right, proving that the above move was impossible.

In fact, the problem of the seven bridges cannot be solved, but this is the problem of one stroke. There is a pattern to this problem: if the number of bars connected by the starting point to other points is even, you can go back to the original point.

You can't solve it, only even numbers can.

You can't draw it down in one stroke.

Seven bridges are connected.

This is a deceptively simple question, but many people have tried and failed to find the answer. Therefore, a group of university students wrote to the great mathematician Euler, who was only 20 years old at the time, and asked him to analyze it. From the failures of thousands of people, Euler guessed with deep insight that it may not be possible to walk all seven bridges at once without repeating them.

To prove this conjecture correct, Euler used simple geometry to represent land and bridges. He solved the problem in this way: since the land is the connecting point of the bridges, we might as well think of the land separated by the river in the diagram as four points A, B, C, D, and the seven bridges are represented as seven lines connecting these four points.

Odd and even dot plots.

What is an even point? A point is even if it has an even number of edges. Points a, b, e, and f of the "odd and even dot diagram" below. Conversely, if a point has an odd number of sides, it is a singularity. As shown in the figure, C and D.

Do even points and singularities have anything to do with whether or not you can cross the bridge at once? Don't worry, let's take our time.

Euler believes that if a picture can be drawn in one stroke, then there must be a starting point and an end point. The other points on the map are "crossing points" – you have to pass through them when you draw them.

What are the characteristics of "crossing points"? It should be a point of "entering and exiting", and if there is an edge entering this point, then there must be an edge out of this point, and it cannot be that there is no entry and no exit or there is no entry and exit. If there is only in and no out, it is the end; If there is no entry or exit, it is the starting point.

Therefore, the total number of edges in and out of the Crossing Point should be an even number, i.e., the Crossing Point is even.

If the starting point and the end point are the same point, then it is also a point that "enters and exits", so it must be an even point, so that all the points on the graph are even points.

If the start and end points are not the same point, then they must be singularities, so this graph can only have a maximum of two singularities.

To sum up what is said above, it is simply as follows:

There are only two types of shapes that can be drawn in one stroke: one is that all the dots are even. The other type is a graph with only two singularities.

Now compared to the diagram of the seven-bridge problem, let's go back and look at Figure 3, the four points A, B, C, and D are all connected by three sides, which are odd-numbered sides, and there are four in total, so this diagram must not be drawn in one stroke.

Euler's study of the "Seven Bridges Problem" was the beginning of the study of graph theory, and at the same time provided a primary example for the study of topology.

In fact, this kind of one-stroke painting game has been circulating among Chinese folk for a long time, and from long-term practical experience, people know that if all the points of the picture are even points, they can choose a point as the starting point and draw it in one stroke. If it is a figure with two singularities, then choose a singularity as the starting point to finish it smoothly in one stroke. If you don't believe it, you can try the "odd and even dot diagram" in the picture above, choose C and D two singularities to draw, and you will definitely be able to draw it in one stroke.

It's just a pity that for a long time, people have only regarded it as a kind of interesting game, and have not paid attention to it, and no mathematicians have summarized and studied it, which cannot but be said to be a pity.

The Seven Bridges question cannot be drawn in one stroke, and there is no answer to this question.