Math Problem Why does a piece of paper fold in half up to seven times?

One day, I was in a math class and was in a daze as usual. When he was bored, he reached out and picked up the newspaper on the table and began to fold the plane. But he was soon discovered by the teacher.

The teacher did not reproach, but asked, "How many times can a piece of paper be folded in half?" ”。

So I started to try folding the newspaper in half. The teacher immediately stopped me and asked me to take a county estimate. I said it 20 times.

Start folding it right away. But no matter how hard you try, you can't fold the eighth time. I was not convinced

As a result, I was disappointed again, and I had to barely fold it for the eighth time.

The teacher helped me analyze it:

Fold once: thickness 2t, area 1 2t

Fold twice: thickness 4t, area 1 4t

Fold three times: thickness 8t, area 1 8t

Fold four times: thickness 16t, area 1 16t

Fold five times: thickness 32t, area 1 32t

Fold six times: thickness 64t, area 1 64t

Fold seven times: thickness 128t, area 1 128t

Fold eight times: thickness 256t, area 1 256t

Fold nine times: thickness 512t, area 1 512t

It can be seen that the thickness of the newspaper increases in proportion to the number of folds, and its area decreases in this way. Combined with the pull of the paper itself, it is more difficult to fold a newspaper in half 9 times than to fold 512 newspapers in half at once!

ideally, no other factors are taken into account)

After folding seven times, the thickness of the paper is seven times that of the original two.

The area of the surface is one-seventh of the two, and the longitudinal length of the surface is one-sixth and one-half of the square of two.

In other words. Only when the longitudinal length of the paper is 2 times 13 and a half of the thickness can it continue to be folded.

The thickness of a piece of paper is about millimeters.

The thirteenth and one-half square of 2 is 11600

Its length is 116,232 meters.

Obviously, we couldn't have gotten our hands on such a large piece of paper.

Divide the area of the paper by two to the seventh power and see what the result is.

I don't know anything about math, so I can only watch the fun, so I'm okay.

You can fold it in half many times.

First of all, the answer is more than 7 times. The number of times a piece of paper is folded in half depends on the thickness of the paper and the size of the paper. If the paper is thin and large enough, it is fine to fold it in half 12 times.

As far as current technology is concerned, the thickness of paper is sky-high. However, the size of the paper is basically infinite for the time being (because it can be spliced), so the number of folds in half is theoretically infinite.

I try it

In fact, the best way to deal with this problem is to practice it yourself.

Prepare paper of any size, do a folding, I tried it, whether it is a normal thickness of paper, or a napkin, it is indeed after 7 times, it is difficult to fold the eighth time, if not with the help of external forces!

Scientific basis

So what is it? Build a mathematical model.

From a theoretical point of view, when the number of folds n is an even number, the length of the folded edge is l (2 (, the thickness becomes 2 n*h, and it cannot be folded when n>2 3*(log2(l h)-1) is satisfied. According to the general condition of the paper, the thickness is about 1m, and the side length is 1m, according to the above formula, it can be concluded that it cannot be folded when n >. Theoretically, the limit number should be eight.

The rumors are over

1. External force assistance.

In fact, with the help of external forces, it is possible to break through this limit, for example, there is a guy abroad who successfully reached 8 times with the help of a press.

Second, oversized paper.

There is also a team abroad, using oversized paper, which has successfully reached 11 times.

Personal opinion

A plain sheet of printed A4 paper.

The thickness of the general is.

Fold in half 3 times for the thickness of your nails.

Fold in half 7 times The thickness of a 128-page notebook.

Fold in half 10 times the width of one hand.

Fold in half 23 times for one kilometer, about 3,280 steps.

LordThe reason for this is that the theory is built on exponential growth, as if you were holding a giant engineering computer and pressing 2x2x2x2 all the time....until 23 times. The rate of growth in numbers is staggering.

Because the highest limit of a piece of paper is to fold it in half 7 times, and by the time it is folded in half for the 7th time, the paper can no longer be bent.

Say a piece of paper, fold it in half no more than seven times, because if you fold it in half once, the thickness becomes twice as much, and the area is 1 2, and the thickness is 2 when folded once, and the area is 1 2;Fold twice with a thickness of 4, an area of 1 4;And so on, the thickness of the seven folds is 128, and the area is 1 128.

Because after folding it in half several times, the paper becomes very hard, and it is more difficult to fold it in half after it is hard, and it will not be able to fold.

There is no reason. Fold only seven times?

When most people fold paper, they will say that a piece of paper can only be folded in half 7 times. To be precise, there is no basis for this claim. It's just that as the paper is folded in half, the thickness and the amount of force required increases, and it becomes difficult to fold it in half again.

The difficulty definitely increases as the number of times increases, but seven times is definitely not the limit of a piece of paper. We need to take into account the thickness of the paper, its length, and the maximum amount of force we can give.

1. Seven times is just a symbol No matter in theory or practice, seven times is definitely not the limit of a piece of paper, as long as you have enough ability, you can definitely exceed seven times. However, I personally believe that the statement of seven times is not wrong, it symbolizes a difficulty and height. It is to tell people that when you fold it in half seven times, it is difficult enough, and you have to use more force to fold it in half again.

After all, every time you fold in half, the thickness of the paper will double the current one, and the size will be doubled in size, and by the seventh time, the thickness and size will also reach a certain point.

2. Being able to break through seven times seven times is a symbol, and it is very simple to break through it. There are also all kinds of weird experiments on the Internet, such as using a roll of toilet paper, spreading it all out and folding it in half. The length of a roll of toilet paper is definitely enough, it depends on the thickness and strength.

As long as you have enough strength and height enough to fold in half, you can definitely break through seven times. The most I saw was 9 folds in half, also on toilet paper. However, this only represents an individual, and it cannot be said that all people who use toilet paper can be folded in half nine times or cannot break through nine times.

With the help of some props, you may be able to break through nine times.

3. In fact, there are many sayings in life, which seem to be very correct, but in fact it is a symbol. It only tells you that this thing is difficult, but it doesn't mean it's the truth. As long as you are willing to try, breaking through the existence of symbolism is a very simple thing.

Because that's the truth, there's simply no paper that is folded in half more than seven times, and if there's not enough paper, you can take a piece of paper and experiment with it, and the paper will become so small that it can't be folded in half.

For the average person, there is no basis for the statement that a piece of paper can only be folded in half seven times, and the difficulty will definitely increase as the number of times increases.

will exceed its own limits. Although it can be calculated more than 7 times, it is difficult to exceed its own limit in real life.

Is it true that a piece of paper can only be folded in half 7 times? Is the larger the paper, the more often it is folded in half?

This is just a summary of a small phenomenon in life, he can't talk about any scientific truth, just like saying that others tell you that you drink 500 ml of water at a time is about the same, you have to drink more, then of course it is okay, you drink a liter of water at a time and no one cares about you, this is just a matter of common sense,There is no question of who is right and who is wrong, only if you can come up with another counterexample to prove that his example is wrong, then your example is right.

A piece of paper is thin, but folding it in half can make it exponentially thicker. Fold in half once is two layers, twice 4 layers, three times 8 layers, seven times is 128 layers, for the size of the paper we usually use, and then want to fold down with the help of a hydraulic press. However, if you choose a larger and thinner paper, you can fold it more than seven times.

If the thickness of the paper itself is a, the thickness of the paper itself is 128a, and the paper itself is not absolutely folded in half, and after seven times, the inner folds and the thickness of the paper together will break the outer paper.

According to the knowledge of physics, it is known that a piece of paper folded in half 7 times will reach a thickness equivalent to the very thickness of the paper itself, which can be done in theory, but in practice it cannot be done.

OK. As long as the paper is large, it is okay.

8. The ninth.

A piece of paper can be folded in half a few times at most, which seems like a boring problem. You might say that if you give me a piece of paper that's big and thin enough, I can fold it 100 million times. This is true, theoretically it can be folded countless times.

But in real life, if you take a piece of paper and test it yourself, you will be surprised to find that it is generally difficult to exceed 7 times, and at most 8 or 9 times. It is said that the most recent world record is 12 times (that is, the supergirl in **). One can't help but ask, why is it so difficult to fold a piece of paper in half after only a few times?

Let's analyze it: 1) Each fold must be folded in half with the radius of the previous thickness, which needs to consume the length or width of the paper. 2) Any material is elastic when bending, and when the thickness reaches a certain level, it needs a certain length to be folded in half, otherwise it will be broken.

3) Paper folded n times in half is much more elastic than paper that is simply stacked with the same number of layers, and its thickness cannot increase in a way that theoretically 2 to the nth power. Therefore, after a certain thickness, it is very difficult for human hands to fold. 4) The one-way folding formula is deduced through practical verification, the diameter of the unit circle is , and the length consumed by folding a piece of paper once (folded into a unit circle) is set by using the limit method (the diameter of the unit circle is a benchmark unit q(q=).

n is the number of folds, and l is the length of the paper consumed. According to the following reasoning, the formula for one-way folding of origami can be obtained: l= +4)*(2 n-1) 6 q 2q 3q 4q 5q 6q 7q 8q 9q 10q ...

16q 17q ..32q 33q...64q...

n=1 1 n=2 2 1 n=3 3 2 1 n=4 4 3 2 2 1 1 1 1 n=5 5 4 3 3 2 2 2 2 1 1 ..1 n=6 6 5 4 4 3 3 3 3 2 2 ..2 1 ..

1 n=7 7 6 5 5 4 4 4 4 3 3 ..3 2 ..2 1 ..

1 ..l = If a thick piece of paper is folded in half 21 times, then its thickness reaches more than 100,000 kilometres (the distance between the earth and the sun is tens of thousands of kilometres). It's hard to imagine that a thin piece of paper can reach such an astonishing thickness after only 20 folds.

There are many little-known things in life that are actually all around us, and as long as you are good at observing and analyzing, the fun is naturally wonderful.

Every time you fold it, the width of the paper is halved, but the thickness is doubled After seven folds, the width of the paper is only 1 128, and the thickness is 128 times that of the original At this time, the width of the general paper may be less than the total thickness, making it difficult for you to fold again I heard that some people find a particularly thin paper that can be folded eight times! The way we fold will also have an impact, usually we will rotate the paper 90 degrees each time we fold again, which will make it more difficult to fold in the future, if we fold in the same direction, the result will be different Just think about pulling a roll of toilet paper apart, folding it in the same direction only, folding more than 7 times! It's fun!

The relationship between the number of layers of paper a and the number of folds n is: a = 2 n, which means that the non-fold is 1 layer of 1 layer of half folding, 1 time is 2 layers of half folding, 2 times is 4 layers of half folding, 3 times is 8 layers of ......Fold it in half 7 times and there are already 128 layers of paper, the paper is very thin, but after all, there is a certain thickness and then fold it in half again is 256 layers, when 256 layers, the fibers in the paper on the outside can no longer be destroyed to produce creases, you can imagine the situation where 128 sheets of paper are folded together, 128 sheets of ordinary paper (equivalent to a book of 256 pages) are already very thick (about 1 cm), and it is very difficult to fold the whole book in half, which can be imagined.

Fold once for 2 layers and twice for 4 layers. According to the exponential law, there are 128 layers in 7 times, is there such a large paper? So no!

It seems that it can't be so thick that it won't break in the end.

Can't you? “？/

Can you fold it 7 times?

It's not that you can't fold, if the paper is big enough, do your own calculations, one square meter of paper divided by two, after 7 consecutive times, there is still the size of a palm, the thickness becomes 128 times, if the strength is big enough, for example, there is a hydraulic press is completely possible, if the paper is large enough, ten times is also possible.