Some time ago, I saw Kathy studying a stroke in the exploration course and suddenly became interested.
I checked Baidu and found that it is an important question type in Olympic Mathematics.
Stroke belongs to the category of graph theory and geometric topology in mathematics. I haven't understood the mathematical description of stroke on the Internet for a long time.
Calm down, make a cup of hot tea and listen to iced coke. After the baptism of ice and fire, my mind is much clearer! Take out the pen and paper, and after an hour of groping, I finally understand how to solve the problem. Let's share it:
To finish a stroke, you need to do two things:
1, how to judge whether a painting can be drawn with one stroke?
2. How to draw it with one stroke?
We understand the problem-solving process in detail through three examples (representing three situations):
Example 1 Problem solving process:
The number of edges of each point (called degree in graph theory) is odd and the number is 0 (the number of edges of each point above is even, so it is even, and there is no singularity).
Remember the main points:
If the odd number is 0, there must be a sum.
When drawing, you can start from any point and end at that point to form a connected graph.
Remarks: Figure B is to demonstrate whether the intersection point should be marked, because singularity is the key to judge the existence of strokes, and if two lines intersect and are even points, it is not important whether the intersection point is marked. If the intersection is a singularity, it must be marked.
For example, as shown in the following figure: 0/ 1/2/3 is the intersection point, but the degrees are all 3, which are singular points and must be marked.
The above demonstrates that the number of singularities is 0, there must be strokes, and drawing can start from any point.
The second case of stroke is as follows:
Example 2 Problem solving process:
The odd number of the edge of each point (called the degree in graph theory) is 2([ 1:3] and [4:3]).
Remember the main points:
If the odd number is 2, there must be a sum.
When drawing, you must start from a singularity and end with another singularity.
Circuit diagram from singularity 4 to singularity 1 end.
Circuit diagram from singularity 1 to singularity 4 [a-b-c-d-e-f-g-h-i-j-k]
There should be other solutions, so I leave them to you to think about. You can try with a pen, hehe!
Remember the main points:
If the number of singularities is not 0 or 2, surely * * * can't draw a stroke! ! !
Let's make sure
Example 3:
Let's use the simplest domain vocabulary to demonstrate:
Let's modify the diagram in Example 2 and see what happens.
At this point, the key points of a pen (whether it exists and how to draw it) will be made clear. But in fact, we also include a condition that is not mentioned: the graph must be connected.
Give a simple example: go back to this picture, you can never draw it in one stroke, because there is no connection between the inside and the outside.
So to sum up the main points of stroke:
1, must be a connected graph.
2. The number of singularities must be 0 or 2, otherwise it is definitely impossible to draw a stroke.
3. If the number of singularities is 0, when drawing * * *, it starts from any even point and ends at that point * * *.
4. If the number of singularities is 2, then when drawing, it starts from any singularity and ends at another singularity.
You can practice more complex strokes online, and with these foundations, it will be relatively easy to understand.
However, some complex drawings that meet the above conditions are still very brain-burning. Good luck to you and your children!
Reference description:
Some pictures are quoted from/news/20120808/59501.html.