Yu Xia: So math can be learned like this.

I find myself sometimes inclined to read "Ye Gong loves dragons".

Every time I hear of a good book, I always think silently and whisper, hoping to read it quickly. But when there are books, they are often put into the "cold palace" for many reasons, and no one cares.

Fortunately, this book can't speak, otherwise it will ridicule me as a "fake reader".

No, at the end of last year, my friend took the trouble to find three copies of Liu Xunyu's Mathematics Can Be Learned in this Way from the Internet for me. Turn over a few pages and you will see n! There are symbols such as sigma, and I suddenly felt numb in my scalp, and then my old illness recurred, leaving it in line with many books.

Although things have been going on these days, there is still a lot of time to maneuver. A few days ago, I gave myself a serious self-criticism. So, like a strong man's broken wrist, I took out the series "Interesting Mathematics" of "Mathematics can be learned in this way" and finished it in three days.

I finally finished reading it today. If you ask me, how do you feel? I tell you, I want to finish the other two books quickly.

The charm of this book is so great. Who is the author Liu Xunyu?

Mr. Liu Xunyu is a famous mathematics educator in China. His educational career spanned two periods: the Republic of China and New China. He has served as a math teacher or principal in many universities and middle schools, and served as deputy editor-in-chief of People's Education Publishing House. The national mathematics textbooks for primary and secondary schools have been approved, a large number of mathematics education papers have been published, and many mathematics textbooks and popular science books for primary and secondary schools have been published.

1983, when Yang Zhenning introduced Mr. Liu Xunyu's mathematics learning process to middle school students in Hong Kong, he specifically mentioned him. He said: "There is a Mr. Liu Xunyu, who is a mathematician and has written many simple and interesting mathematical articles. I remember reading one of his articles about intelligence tests before I knew the extremely important mathematical concepts of permutation and parity. "

The famous scientist who won the Nobel Prize in Physics praised him so much, which shows how powerful Liu Jiaoshou is.

To my surprise, such a popular math book was prefaced by Mr. Feng Zikai, a famous writer and painter. Crossing the border is quite fashionable.

Mr. Feng Zikai said: "I have never tasted the interest of mathematics, nor have I visited the world of mathematics. Thanks! It is the recent articles in this book that have slightly compensated me for this loss. After I met Yu Xun, he wrote these articles. I read it every time he publishes it. What attracted me to read it was their interesting themes. I am often unconsciously lured into the world of mathematics. "

It's hard to understand how interesting it is without being there. Therefore, I have to choose some interesting questions for everyone to try.

First of all, the last "Han Xin points soldiers"

When Professor Liu Xunyu said that he was a primary school student, a salt boss gave him a test and said that he would invite him to dinner.

The topic is clear as soon as it is said.

Two of the three plots remain, three of the five plots remain, and two of the seven plots remain. How many?

At that time, Liu Xunyu was also high-spirited and full of ambition, thinking, isn't this a common multiple? One is in pediatrics.

So, he hurriedly said, forget it, there is more than one.

Boss Yan even praised his children for their cleverness.

Liu Xunyu thought of the topic he had done before. There are two differences in three places, four differences in five places and six differences in seven places, at least a few. Is to find the least common multiple of 3, 5 and 7, and then add 1 (exactly the remainder 1) to get 106.

So, he worked out the answer with his own set of ideas, the lowest is 104, and there are 209.

As a result, boss Yan said it was wrong. I checked, but it wasn't.

After Liu Xunyu came back, he was scolded by his grandfather and warned him that "I would rather not be in front of people than be incomplete."

If he was really a grandfather in feudal times, our parents and friends today would certainly not accuse their children like this. They will definitely ask the salt boss how to work out the correct answer.

So how to solve this problem?

This question comes from the calculation of the mathematical classic Sun Tzu's Art of War. "Some things are unknowable. There are two left in three or three, three left in five or five, and two left in seven or seven. Ask geometry? "

Later, in order to make this problem more specific, people adapted it into the problem of "Han Xin ordering soldiers".

After a battle, Han Xin wanted to count the number of soldiers. Let the soldiers work in groups of three, not two; A group of five people, not a group of three; Seven people in a group, two people can't be grouped. How many soldiers are there?

What do you think? Let's remember two common sense:

First, whether the multiple of a certain number is a multiple of a certain number;

Second, the sum of several multiples of a number is still a multiple of a number.

35 is a multiple of 5 and 7. Divide by 3 to get 2.

2 1 is a multiple of 3 and 7, divided by 5 remainder 1. If we want to keep 3, we should include three 2 1, that is, 2 1×3.

15 is a multiple of 3 and 5, and dividing by 7 equals 1. If you want to make 2, you must include two 15, that is, 15×2.

Add up the above three figures.

35+2 1×3+ 15×2= 128。

105 is a common multiple of 3, 5 and 7, so the remainder divided by 3, 5 and 7 will not change after adding or subtracting 105. 128- 105=23. The general solution is: 23+ 105n, where n = 0, 1, 2,3 …

Is it hard to understand? Then look at Artu's demonstration method: (from top to bottom)

35 + 2 1 ×3? + ? 15×2? = 128

A multiple of 7? +? A multiple of 7? +? The remainder of 7 is 2 = the remainder of 7 is 2.

A multiple of 5? +? A multiple of 5 and a remainder of 3+? Multiply of 5 = Multiply of 5 and remainder of 3.

A multiple of 3, a multiple of 2+3? +? Multiply of 3 = Multiply of 3 and remainder of 2.

Understand? Ok, let's have a try.

More than three in four plots, two in five plots and three in seven plots. What is the minimum quantity?

Is it a little round? Headaches are not bad for me.

Let's go somewhere else to watch Luohan Dui.

What do you mean by piling arhats? Counting from the bottom row, there is at least one person in each row until there is only one person at the top.

Mathematically, we call a group of numbers with the same difference as arithmetic progression. It is not difficult to understand arithmetic progression's calculations, such as

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

Anyone who has heard the story of Gauss knows that the beginning and the end are added, multiplied by the number of terms, and then divided by 2. Expressed in letters: n(n+ 1)/2.

Similar to this property, there are the sum of squares and cubes of integers from 1 to a certain number:

1^2+2^2+3^2+4^2+5^2……

1^3+2^3+3^3+4^3+5^3……

(2,3 is the format of square and cube)

Do you remember the summation formula? They are:

∑n^2=n(n+ 1)(2n+ 1)/6

∑n^3=[(n×(n+ 1))/2]^2

But how is it derived? Mr. Liu Xunyu's method is wonderful.

Find the sum of squares:

The square of 1, 2, 3, 4 can be represented by a small square.

1^2+2^2+3^2+4^2

Pile them up, and they will look like figure 1 or figure 2. Combine the two graphs of 1 with the second graph to form the third graph, which is three times the sum of them.

The length of the third graph is 1+2+3+4, and the width is 2× 4+ 1.

Because 1 2+2 2+3 2+4 2 is one third of their area, the sum of squares is:

4(4+ 1)/2? ×( 2×4+ 1)÷3

Change it to the general law of n-push.

n(n+ 1)(2n+ 1)/6

Of course, this inductive method is not strict. We need to use n+ 1 again and apply it to the formula. If it doesn't hold, we don't need it here. Everyone knows anyway.

So how do you find the cubic sum?

Please look at the picture below. Did you find that the cube of 2 is the square of 3 minus the square of 1? The cube of 3 is the square of 6 minus the square of 3, and the cube of 4 is the square of 10 minus the square of 6. Put them together, it is exactly the square of 10.

So the square of 1 3+2 3+3 3+4 3 = (1+2+3+4).

Push it to the general law, that is

[(n×(n+ 1))/2]^2

Mr. Hua once said:

It is not intuitive when the number is missing, and it is difficult to be nuanced when the number is missing.

Numbers and shapes are well combined, and everything is separated.

Do you think it is particularly accurate to use this poem to express your feelings at this time?

Ok, let's practice one more question.

1/2+ 1/4+ 1/8+ 1/ 16+ 1/32+ 1/64+ 1/ 128=

You might as well draw a picture, and you will get something.

Are you tired? Over and over again, add more oil and go on.

Let's take a look at Eight Immortals Crossing the Ocean.

I wonder if you have ever encountered something like "Eight Immortals Crossing the Sea"? I have seen it in some tourist attractions. For example, fortune tellers take out hundreds of surnames and ask people to open them. After clicking, they can guess what other people's surnames are. It's a little complicated. Say it simply.

A person arranges eight different kinds of money in two rows on the table, telling you to look at one and keep it in mind.

He put the money away, rearranged it, or two rows up and down. He also showed you which line you recognized last time, remember.

He put the money away again and arranged it in two rows again. This time, he asked you to see it and tell him the location of the money you saw.

For example, if you say "up and down" to him, he will show you the second one in the next row. Although you feel a little strange and want to deny it, your face will not hide it for you.

Why does this man have such ability? You may suspect that he guessed by accident, but he will never fail once, twice or thrice, which is certainly not accidental.

What is the secret here? For convenience, I put these eight money in letters. Let me put it this way first:

DCBA

HGFE

You said, then, it must be ABCD. He put it like this, both left and right, in no particular order.

OOCA

Object-oriented database

Then you said the next one, it must be BD, and he did it again. Other order is not important.

OOOB

OOOD

If you say it again, it must be B.

Of course, it's naive. Now let's have an upgraded version.

It is someone who watches you swing three times before telling you to go up and down, so you can know which letter he is thinking.

Really? Just try it.

Let me put it this way first:

D C B A

H G F E

Then from right to left, one by one.

(AEBFCGDH), and then from left to right, put down a row first, and then put on a row.

Federal Bureau of Building and Engineering

High density graphite carbon

Also from right to left, one by one collection (ACEGBDFH).

Or from left to right, put down a row first, and then put on a row.

ge

h？ Food and drug administration

Why can you guess? Look at the position of these letters.

A up, up, up, b up, down.

C up and down? Up and down

E, up and down Up and down

G down, up, down? H Xia Xia Xia

All eight letters are in different positions. He can't say which one, but you can point it out.

Do you understand now? When the Eight Immortals crossed the sea, what was hidden was the arrangement. There are many arrangement problems in life.

For example, a square table, now there are eight guests coming to your house. So, how many different arrangements are there for their seats?

It is usually thought that there may be dozens.

Liu Jiaoshou asked us to fix a position first, and let eight people take turns to sit in this position. After the first person is fixed, there are seven people in the second position who can take turns sitting. And so on, namely:

8X6x6 X5x3x2x 1 = 40320 (type)

This way of speaking is estimated that many people still feel foggy, so let's go back to the original point and think about it:

1 a person sitting in a position is 1 a situation.

Two people sit in two positions, namely 12 and 2 1.

Three people sit in three positions, namely 123,132,213,231,312,321.

Four people sit in four positions, namely 1234, 1243, 1324, 1342, 1423, 1432, 2/kloc-0. 3 124, 3 142, 32 14, 324 1, 34 12, 342 1, 4 123, 4 132, 4265438.

If I continue to go public, I will faint. Did you find a pattern?

1! = 1

2! =2x 1= 2

3! =3x2x 1=6

4! =4x3x2x 1=24

In a word, the arrangement of all n non-repetitive things is the factorial of n.

All right, let's make a change. If there are 18 players, what will happen if you choose 1 1 person to participate in the competition?

I am especially willing to take a step back when I encounter difficulties. I think it is of great benefit to solve problems and life.

As the enlightenment poem of the cloth bag monk said:

Insert the young crops into the field with your hands, and you can see the water in the sky with your head down;

A pure heart is the way, and retrogression is the progress.

Let's think about it first:

If there are four people, choose two people to compete, what are the situations?

Use 1, 2, 3, 4 to represent these four people.

If it consists of two digits, it is12,21,13,31,14,41,23,32,24,42,34.

But the trials, 12 and 2 1, 13 and 3 1, are all the same. therefore

4x3÷( 1 x2)=6

If four people choose three people to compete, what are the situations?

If it consists of three digits, there are 24 situations:

1 There are six kinds of kings (one hundred years):

123, 132, 124, 142, 134. 143

There must be six other kings, so there are 24.

But 123, 132, 2 13, 23 1, 3 12, 32 1, are all 1, 2, 3? So divide by 6. There are six ways to put three digits in three digits, except 0. )

4x3x2 ÷(3x2 x 1)=4

Choose m people from n people to participate in the competition, which is n! ÷m！

Try again?

There are five different letters. Choose three to play. How many different ways are there?

My friend, if you see this line, it means that you really love mathematics; If you not only understand, but also understand, it means that you are really good at math; If you don't insist on seeing this place, it means that your talent may be outside of science. ...

Haha, I'm just playing with you. Exercise your brain and be happy! As the famous mathematician Mr. Chen Shengshen said: Mathematics is interesting.

Finally, I quote Russell: Mathematics is such a thing that people who study it don't know what they are doing.