Residents of this city often walk across the bridge along the river. There is a young man in the city who is very smart and loves to think. One day, the young man asked everyone this question: Can you cross seven bridges at a time, each bridge can only cross once, and finally return to the starting point? This is the world-famous seven-bridge problem, and people at that time never found the answer.
Euler, a great mathematician, heard this question from a friend and soon proved that such a move did not exist. Euler solved the problem in this way: the land separated by rivers in the map is regarded as four points A, B, C and D4, and the seven bridges are represented as seven lines connecting these four points. The thinking process is as follows:
The great mathematician Euler cleverly abstracted such a practical problem into a simple geometric figure composed of points and lines, and transformed the problem to be solved into a problem in Figure 2. Such an abstract process is the most wonderful thinking of Euler when solving this problem, and it is also the most worth learning. Because Figure (2) can't be drawn in one stroke, it is impossible for people to cross seven bridges at once. 1736, Euler published the results of this problem in the Journal of St. Petersburg Academy of Sciences. Euler's research on "Seven Bridges" is the beginning of graph theory research. It can be said that it is this research that makes it the originator of graph theory.
So how did Euler judge that Figure (2) could not be drawn with one stroke? In order to make it easier for everyone to understand, combined with this example, I use my own language to explain the idea of solving a stroke problem: this figure has 4 points and 7 lines, and each point is the common endpoint of several routes. If a point is the common endpoint of even lines, we call it an even point (or even point); If a point is the common endpoint of odd lines, we call it an odd point (or singularity). Point A in Figure (2) is the common endpoint of five lines, and points B, C and D are the common endpoints of three lines, so Figure (2) has four singularities. Generally, we call the starting point and the ending point as passing points. Obviously, if there are incoming lines, all passing points in the stroke graph must have outgoing lines, so each point must have even lines to complete a stroke. If there is a singularity at the passing point, there will inevitably be unpaved routes or repeated routes. Therefore, in a stroke diagram, only the starting point and the ending point can be singular points (the starting point can only not enter, and the ending point can not enter this point at last), that is, there can only be two singularities at most, one singularity as the starting point and the other singularity as the ending point. Because there are four singularities in Figure (2), Figure (2) cannot be drawn with one stroke.
Two other points:
First of all, all lines in the drawing must be continuous, because the pen will not leave the paper. If a picture is composed of two unconnected parts, it is definitely impossible to draw strokes. For example, the word "country" cannot be written in one stroke.
Second, the singularities in the stroke graph all appear in pairs (because each line has two endpoints, and the sum of the endpoints of all lines is even). If there are no singularities in the graph, they are all even points, which can be drawn in one stroke, but the starting point and the ending point must be the same point.
Combining the above description, to solve a problem, the first step is to find out all the points in the graph and judge whether they are singular points or even points; The second step is to make a correct judgment according to the number of singular points; The third step is to let the children try to draw a picture with a pencil to verify their judgment.