Seven Bridges Problem 65438+One of the famous classical mathematical problems in the 8th century. In a park in Konigsberg, there are seven bridges connecting two islands and islands in the Fritz fritz pregl River with the river bank (pictured). Is it possible to start from any of these four places, cross each bridge only once, and then return to the starting point? Euler studied and solved this problem in 1736. He simplified the problem to the "one stroke" problem shown on the right, which proved that the above method was impossible. Hot issues in graph theory research. 65438+ K? nigsberg, Prussia. At the beginning of the 8th century, the Fritz fritz pregl River passed through this town. Naif Island is located in the river, and there are 7 bridges on the river, connecting the whole town. Local residents are keen on a difficult problem: is there a route that can cross seven bridges without repetition? This is the problem of the seventh bridge in Konigsberg. Euler used points to represent islands and land, and the connecting line between the two points represented the bridge connecting them, which simplified rivers, islands and bridges into a network, and turned the problem of seven bridges into a problem of judging whether the connected networks could draw a sum. He not only solved this problem, but also gave the necessary and sufficient conditions for connected networks to be brushes, if they are connected and the odd vertices (the number of arcs passing through this point is odd) are 0 or 2. When Euler visited Konigsberg, Prussia (now Kaliningrad, Russia) in 1736, he found that local citizens were engaged in a very interesting pastime. In konigsberg, a river called Pregel runs through it. This interesting pastime is to walk across all seven bridges on Saturday. Each bridge can only cross once, and the starting point and the ending point must be the same place. Euler regarded each piece of land as a point, and the bridge connecting the two pieces of land represented the famous mathematician Euler with lines.

. It was later inferred that such a move was impossible. His argument is this: In addition to the starting point, every time a person enters a piece of land (or point) from one bridge, he or she also leaves the point from another bridge. So every time you pass a point, two bridges (or lines) are counted, and the line leaving from the starting point and the line finally returning to the starting point are also counted, so the number of bridges connecting each piece of land and other places must be even. The graph formed by the seven bridges does not contain even numbers, so the above tasks cannot be completed. Euler's consideration is very important and ingenious, which embodies the uniqueness of mathematicians in dealing with practical problems-abstracting a practical problem into a suitable "mathematical model". This research method is called "mathematical model method". You don't need to use profound theories, thinking is the key to solving problems. Next, based on a theorem in the network, Euler quickly judged that it was impossible not to visit the seven bridges in Konigsberg at one time. In other words, for many years, the non-repetitive route that people have worked so hard to find simply does not exist. A question that stumped so many people turned out to be such an unexpected answer!

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After the question was put forward, many people were very interested and tried it one after another, but for a long time, it was never solved. And with ordinary mathematics knowledge, each bridge walks once, so all the walks of these seven bridges are 7! =5040 kinds, and in so many cases, it will take a lot of time to test one by one. But how can we find a route and successfully cross each bridge without repeating it? Thus, the famous "Seven Bridges of Konigsberg" was formed. 1735, several college students wrote to Euler, a talented mathematician who worked at the Academy of Sciences in Petersburg, Russia, and asked him to help solve this problem. Euler personally observed the Seven Bridges in Konigsberg, and seriously thought about the way to go, but he never succeeded, so he doubted whether the problem of the Seven Bridges was inherently unsolvable. 1736, 29-year-old Euler submitted his paper "Seven Bridges in Konigsberg" after a year's research, which successfully solved this problem and created a new branch of mathematics-graph theory. In this paper, Euler abstracted the problem of seven bridges, and regarded each land as a point, and the bridge connecting two lands was represented by a line. So as to obtain the geometric figure as shown in the figure. If we use four points A, B, C and D to represent the four regions of Konigsberg. In this way, the famous "seven bridges problem" is transformed into a question of whether these seven lines can be drawn with non-repetitive strokes. If it can be drawn, there must be an end point and a starting point in the figure, and the starting point and the end point should be the same. Because of symmetry, starting with a or c has the same effect. If A is assumed to be the starting point and the ending point, there must be a starting line and a corresponding entering line. If we define the number of rows entering A as in-degree, the number of rows leaving A as out-degree, and the number of rows related to A as A-degree, then A's out-degree and in-degree are equal. That is to say, if there is a solution from A, then the degree of A should be even, but in fact the degree of A is 3 and odd, so we can see that there is no solution from A. At the same time, if we start from B or D, because the degrees of B and D are 5 and 3 respectively, they are all odd, that is to say, there is no solution from them. Based on the above reasons, we can see that the abstract mathematical problem has no solution, that is, the "seven-bridge problem" also has no solution. From this, we get: Euler path relation. From this, we know that the following two conditions must be met if a graph can draw strokes: 1. Graphics must be connected. 2. The number of "singularities" on the way is 0 or 2. We can also use this to test whether the figure can be drawn with one stroke. Looking back, we can also judge the "seven bridges problem" from this. These four points are singularities, so we can see that the figure cannot be drawn in one stroke, which means there is no repetition through all seven bridges. Euler's consideration is very important and ingenious, which embodies the uniqueness of mathematicians in dealing with practical problems-abstracting a practical problem into a suitable "mathematical model". This research method is called "mathematical model method". You don't need to use profound theories, thinking is the key to solving problems. 1736, Euler reported the problem of seven bridges in his paper "Seven Bridges in Konigsberg" submitted to the Academy of Sciences in Petersburg.

In the report, he expounded his method to solve the problem. His ingenious solution laid the foundation for the establishment of a new branch of mathematics-topology. Seven Bridges and euler theorem. Through the study of seven bridges, Euler not only satisfactorily answered the questions raised by the residents of Konigsberg, but also obtained and proved three more extensive conclusions about a pen, namely the geography of Kaliningrad.

Euler theorem. For connected graphs, the route from a node is usually called Euler path. People usually call the Euler path back to the starting point the Euler path. Graphs with Euler paths are called Euler graphs.