A piece of paper cannot be folded nine times.

1. Is it possible to fold a piece of paper nine times and once, and a piece of paper becomes two layers? Fold it again and change it to 4 layers; Fold it in half 3 times and change it to 8 layers ... When the number of folds is n, the paper has 2 n layers.

After being folded in half for 7 times, * * * has 128 layers of paper, which can barely be folded in half. But after eight times, there are 256 layers, and folding in half once is equivalent to folding 256 sheets of paper at the same time, which is extremely difficult.

You can try to fold a book with more than 500 pages (250 pages) and 250 pages (125 pages) in half.

By the eighth folding, the paper has become a cuboid with a side length of about 6 cm and a thickness (height) of about 3 cm. By the ninth fold, the thickness exceeds the side length. No wonder it can't be folded any more.

This machine can only be folded nine times.

Just do the math. If the thickness of the paper reaches half of the folding surface, it is difficult to fold, so it can be inferred that if the paper is square, its side length is a and its thickness is h, when it is folded once, its side length is unchanged, and its thickness is 2 times h, when it is folded twice, its side length is half of the original side length, and its thickness becomes 4 times h, among which it is also folded, and a formula can be deduced: when the number of folds is even, it is 2/3. According to the general paper situation, when the thickness is about 0. 1mm and the side length is 1m, according to the above formula, n >:8. 19 18 cannot be folded, that is to say, the thickness is about 0. 1mm and the side length is/. When considering a large piece of paper with constant thickness and side length 1Km, according to the above formula, N >；; 14.8357 cannot be folded, that is, 14 times can only be folded. So the number of folds is related to the value of l/h. If L/H is infinite, then its logarithm is infinite, and the number of natural folds is infinite. Of course, these are all theoretical conclusions. As for whether or not such a large piece of paper can be folded, how to fold it is impossible to demonstrate.

One last question, if a piece of paper with 1 mm is folded 100 times, its thickness can be calculated as 2 100 * 0.005438+0 m = 65438+65438.5008808886

Theoretically, if the thickness of paper is zero, it can be folded in half countless times. However, due to the actual thickness of the paper, this theory does not exist, because the width of the folded paper cannot be less than or equal to the thickness of the paper, that is, a piece of paper with a thickness of 1 mm should be greater than1mm.

So how many times a piece of paper can be folded in half at most is actually a variable, depending on the actual thickness and size of the paper. Folding a piece of paper with a thickness of 1mm in half for 100 times can exceed the distance from the earth to the moon, which is just an unrealistic mathematical theoretical reasoning diagram.

According to the actual calculation, the base paper size of the new board is 840mm* 1 188mm (this is the size of 16 A4 sheets). If the paper thickness is 1mm, the size of the paper folded in half 1 time should be 840mm*593.5mm (including 0.5). The actual size of the two folds is 593.5mm*4 19.5mm, and the size of the three folds is 295.75mm*4 19.5mm, that is, the actual size after each fold is subtracted from the thickness loss of the folded edge (of course, if it is not folded in half, but cut, this loss can be omitted). The paper size after four folds should be 2000. Theoretically, the size of the paper should be 3.28125mm * 3.330625mm * 3.330625mm when it is folded for the 16th time (not including the folding loss), but if the folding loss is calculated, it can only be folded for the 12th time.

Some classical geometric drawing problems, such as bisection of an angle or doubling the volume of a cube, have been proved to be unable to be solved by drawing with a ruler. But it can be solved by several origami steps. Generally speaking, origami can solve algebraic equations no more than 4 times by drawing. Huzita-Hatori axiom set is an important research achievement in this field.

As a result of studying origami by using geometric concepts, Haga theorem can be used to accurately divide one side of paper into three, five, seven and nine equal parts. Other theorems allow us to fold other patterns from squares, such as equilateral triangles, regular hexagons, regular octagons and specific rectangles, such as gold rectangles and silver rectangles.

Marshall Bern and Barry Hayes [1] have proved that the problem of refolding from creased flat paper to its original shape is NP-complete. Other technical achievements are introduced in more detail in the second part of the book Geometric Origami Algorithm. [ 1]

When a piece of paper is folded in half, the loss function is,

Here l represents the minimum length of paper, t represents the thickness of paper, and n represents the number of folds. This function was put forward by Britney Gallivan at 200 1 (when he was a high school student). He can fold a piece of paper in half 12 times. People used to think that no matter how big the paper was, it could only be folded in half eight times at most.

2. A piece of paper can be folded in half several times. Yesterday, Mr. Guo, a math teacher, mysteriously told us, "Prepare some a4 paper, a tissue and a newspaper tonight." We were puzzled, but Mr. Guo smiled and said, "You will know in math class tomorrow."

In math class, we discussed what the teacher asked us to bring these papers. At this moment, Mr. Guo walked into the classroom and said, "Today, let's do an experiment-how many times can a piece of paper be folded in half at most?" As soon as the teacher solved the mystery, the students began to talk about it. Some say it is ten times, some say it is fifteen times, and some say it is twenty times. Teacher Guo asked if the paper was different and the number of folds was different. What is the most frequently folded and what is the least folded? Finally, we all agree that newspapers fold in half the most times and tissues fold in half the least. Because the newspaper has the largest area and the facial tissue has the smallest area, a4 paper is not too big, so the number of folds is between newspaper and facial tissue.

It can be said that there is no evidence, and it must be proved by hands-on experiments. So, I first took a newspaper to do the experiment. "Once, twice, three times ... seven times." When I folded it for the seventh time, I couldn't fold it any more. I thought to myself: how is it possible, how is it possible, how can such a big newspaper only be folded in half seven times? That paper and a4 paper must only be folded in half five or six times.

Why can't a piece of paper be folded in half nine times? That's impossible. Why? When you fold to the eighth page, the number of layers of paper is 256! As you can imagine, the ninth floor ...

That's impossible, why! When you fold to the eighth page, the number of layers of paper is 256! As you can imagine, the ninth floor ...

In the Japanese science and technology program "Galileo Lab", it seems that the fibers of paper overlap, which leads to very tough ——————————————— I thought about it for a day. It is really because of the fiber overlap of paper. The fiber overlap thickness of paper is equal to the toughness of paper multiplied by the ninth power of 2, which makes the fiber overlap of paper reach a very large data, as if a match is easy to break one hundred and difficult to fold. It's still on the tenth floor, and there's no way to fold it.

4. An experimental composition about the durability of a piece of paper. I am just an ordinary piece of paper. Originally, my sister, my father, my mother and my family were very happy, but that happy family was ...

"Students, what we send below is this dictation of new words." The classroom is very quiet, and more than forty students sit neatly in their seats and listen to the teacher. I'm not idle at Master Xiao Ming's desk either. My eyes stayed on more than forty dictation papers on the teacher's desk, and I was looking for my mother. Soon mother was sent back to Xiaoming's desk by the teacher. "ah! Mom ... I'm here ... "Before I finished, Xiaoming shook his head and said with a sigh," Alas ... it's more than 40 points again, and the word 10, I made six mistakes. What a shame! " After that, he grabbed his mother with his hand and threw it into the dustbin. I froze and couldn't speak for a moment. I just felt something spinning in my eyes. Xiao Li, Xiao Ming's deskmate, saw it and urged, "Xiao Ming, you can't do this. How can a piece of paper …" Xiao Li interrupted her before she finished. "Mother-in-law, you girls are really annoying. Xiaoli sighed and ignored him.

Just when I felt sorry for my mother, the classmate behind Xiaoming patted him on the shoulder. Xiao Ming seemed to understand something. He tore off some pieces of paper from his notebook, scribbled a few big characters, drew a few crooked symbols, and laughed while drawing. He may have written some jokes or something. The classmate behind was about to answer when Xiao Ming told him, "Don't forget to tear it up after reading it." As soon as I find him. After a while, the sound of "hissing" came from behind, and my heart followed it, tugging, tugging.

I'm really afraid that one day, my sister and I will be killed. After a while, what I was worried about really happened. Xiao Ming took his sister in white out of her schoolbag, tore it into pieces, and then wrapped it in a thick bag layer by layer, handed it to the students behind, and introduced it as "playing with evil toys." It looks like there's something in it, but in fact there's nothing. At this time, my heart hurts so much that I can't breathe.

I shouted at Xiaoming desperately: "Give it back to my relatives! Return my loved ones! " But he can't hear you.

In the dead of night, I looked at the moon outside the window and thought, what is our paper for? Is our value just for coquetry, for recording jokes, or for spoofing? I remember before leaving the factory, the workers' uncle said to us, "Your lives were given by human beings, and you should repay them. You are the carrier of human information, a document to witness history, and a warehouse to preserve spiritual wealth. " But what about now?

We came to this world with the dream of rewarding and benefiting mankind, but how are we treated? Waste at will. Someone once calculated an account: waste paper accounts for 18% of the total garbage in China. It takes 17 trees and 100 cubic meters of water to produce a ton of paper, and most of this water will be discharged as sewage. It can be seen that paper is closely related to the natural environment. Schools are big users of paper, and an ordinary middle school student uses 474 pieces of white paper a year. Based on this calculation, how many people can be saved for every 10 thousand students? Someone has calculated that a middle school with 36 classes and 40 students in each class will send 15 notices every semester and print 43,200 copies throughout the year. In fact, these notices can be conveyed orally. Well, I don't want to think about it any more. Will tomorrow be the end of my parents and sisters waiting for me?

A few days later, I was made into a propaganda slogan by the school teacher. The slogan is: "save paper, protect the earth, start with me!" " "Although my life is over, I am very happy. Isn't a small habit a big truth? Yes, save paper and protect the earth, starting with me!

5. Why can't a piece of paper be folded nine times? Fold it nine times, and it is already the ninth power of 2, which is 5 16 layers of paper. If you fold it again, it is equivalent to folding 5 16 sheets of paper together. Do you think the tension of the outermost paper can withstand the folding force?

This question involves definition (concept), and there will be different answers based on what is "a piece of paper" and what is "folding".

If "a piece of paper" refers to ordinary writing paper with the size of A4 or so, and "folding" refers to similar manual folding, then the total thickness of this paper after nine folds is 5 12 times that of a single piece of paper, that is, the thickness at this time is much greater than the width (the width has changed to the original 5 12 1), then because this "paper"

However, if the "paper" is very large and the bending characteristics are very good, it is entirely possible to fold the "paper" nine times.

However, I think the questioner should ask about the common writing paper of A4 size, and "folding" means that folding in half is similar to the usual manual operation. If this question is limited to "common use" (papers), there will be no trouble involving definition (concepts)