1736, 29-year-old Euler submitted his paper "Seven Bridges in Konigsberg" to St. Petersburg Academy of Sciences. While answering questions, he created a new branch of mathematics-graph theory and geometric topology, which also opened a new course in the history of mathematics. After the question of the Seven Bridges was raised.
Many people are very interested in this and have carried out experiments in succession, but for a long time, it has never been solved. Through the study of seven bridges, Euler not only satisfactorily answered the questions raised by Konigsberg residents, but also drew and proved three more extensive conclusions about a stroke, which people usually call "euler theorem F".
Brief introduction to the problem
/kloc-At the beginning of the 8th century, there was a river in Konigsberg, Prussia. There were two small islands in the river, and seven bridges connected the two islands with the river bank (as shown in the overview). A man asked a question: How can a pedestrian walk seven bridges at a time, without repetition or omission, and finally return to the starting point?
Later, the great mathematician Euler transformed it into a geometric problem-the stroke problem. He not only solved this problem, but also gave the necessary and sufficient conditions for a connected graph to be a stroke: the number of singularities is either zero or two (if the number of connected points is odd, it is called singularity; If it is even, it is called even point.
If you want to draw a stroke, there must be even points in the middle, that is, where there is a road, there must be a road, and the singularity can only be at both ends. So any figure can be drawn in one stroke, and the singularity either does not exist or is at both ends).
inference method
When Euler visited Konigsberg (now Kaliningrad, Russia) in Prussia on 1736, he found that the local citizens were engaged in a very interesting pastime. There is a river called fritz pregl in Konigsberg.
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 in the same place. Euler regarded each land as a point, and the bridge connecting the two lands was represented by a line.
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, count two bridges (or lines).
The line leaving from the starting point and the line finally returning to the starting point are also considered as two bridges, so the number of bridges connecting each piece of land with other places must be even.