Bridge crossing problem:

Can you bridge (each bridge can only be built once)?

Example:

Imitate this example and judge in turn:

The picture below shows a small river in the country with six bridges built on it. Can you cross all the bridges at once without repeating them?

(Each small bridge is only allowed to walk once at most, and can walk back and forth repeatedly on land)

3. There is a topic in the book Interesting Mathematics written by China mathematician Chen Jingrun to the effect that there is a river in the French capital and there are two small islands in it. People there built 15 bridge to connect two small islands with the river bank. As shown in the picture below, please tell me if it is possible to cross all the bridges at once from any bank to the other side. (Each bridge can only walk once)

The picture below shows a sales hall. Ask the customer if he can go through all the doors at once without repeating them. Please explain your reasons.

If you design another door at the exit of the sales hall in Room 4, so that customers can enter through the entrance and go through all the doors at once, and then start the sales hall from Room 4, where are you going to open another door?

The answer to an exercise

1. Solution: See the figure below.

Bridge crossing problem:

All the bridges

(Each bridge can only walk once)

One-shot question:

Can you finish it in one stroke (pen can't be lifted and can't be repeated)

2. Solution: As shown in the following two pictures, we can see that it is impossible for us to walk all the small bridges at once without repeating them, because there are four singularities in the picture on the right.

3. Solution: Because the number of bridges passing through either of the two islands is even, and the number of bridges passing through either bank on both sides of the strait is odd, this means that there is a road from either bank, so people can reach the other side by walking all the bridges once. Draw a picture and it will be clear at a glance. See the figure below.

Because there are two singularities in the picture, both of which are shores and a stroke.

So people can cross all the bridges, and each bridge only walks once, from one shore to the other.

4. Solution: After entering the sales hall from the entrance, that is, from 1 room, you can't walk through all the doors repeatedly, because although the whole figure (see the figure below) has only two singularities, the point 1 is an even point.

When the exit is in Room 4, if a door is opened between Room 1 and Room 3, you can walk through all the doors from Room 1 without repeating. Because point 1 becomes a singularity, point 4 is still a singularity, and there are only two singularities in the whole figure, so you can enter from room 1 and exit from room 4. See the figure below.