What is the answer to the Hundred Chickens question? Hundred chicken problems

The question of 100 chickens comes from the ancient Chinese arithmetic book "Zhang Qiu Jian Shujing".

The meaning of the title is this: 1 rooster is 5 yuan, 1 hen is 3 yuan, 3 chickens are 1 yuan, and 100 yuan can buy 100 chickens. Q: How many roosters, hens and chicks can I buy?

There are three answers: 4 roosters, 18 hens, and 78 chicks;

8 roosters, 11 hens, 81 chicks; 12 roosters, 4 hens and 84 chicks.

The 100-chicken problem is an indefinite equation.

The solution to the problem of integer solution is as follows: set the rooster x weaving, the hen y, and the chick z. According to the question, the system of equations can be listed: x+y+z=1005x=3y+13z=100

Eliminate z, we get 7x+4y=100, so y=100-7x4=25-7x4. Since y represents the number of hens, it must be a positive integer.

Therefore, it is necessary to get a multiple of 4. Let's write it as: x=4k(k n).

So y=25-7k. Substituting the original equation, we can get z=75+3k. Write the above three formulas together and have:

x=4ky=25-7kz=75+3k

In general, when k takes different values, many different sets of values for x, y, and z can be obtained. However, for the above specific problem, because y n, k can only take three values, so three answers to this question can be obtained.

The C language requires all three types of chickens.

#include

void main(){

int i,j,k;

for(i=1;i<=20;i++)

for(j=1;j<=33;j++)

for(k=3;k<=300;k+=3){printf("%d %d %d",i,j,k);

Dictionary Explanation: Famous arithmetic problems in ancient China. Originally published in "Zhang Qiu Jian's Sutra" Volume 38:

Today, there is a chicken worth five, a chicken mother is worth three, and a chicken chick is worth one. Where you buy 100 chickens for 100 dollars, ask the chickens, mothers, and chicks? "If the number of chickens is x, the number of hens is y, and the number of chicks is z, then an indefinite system of equations can be obtained for rock acres:

x+y+z=100,5x+3y+13z=100。Although the original book lists all three sets of positive integer answers: (4,18,78), (8, 11,81), or sedan (12,4,84), the basis for the solution is not detailed.

In later generations, many people studied this problem and came up with a solution, which is called "Hundred Chicken Technique".

Chicken Weng one, worth five; The hen is one, and it is worth three; Three chicks, worth one; 100 dollars to buy 100 chickens, then Weng, mother, chick geometry?

Translated, it means five dollars for a rooster, three dollars for a hen, and three dollars for a chicken, and now you have to buy a hundred chickens for a hundred dollars, and ask how many roosters, hens, and chicks are there?

Topic analysis. If you use mathematical methods to solve the problem of buying a hundred chickens for 100 dollars, you can abstract the problem into a system of equations. Let the rooster x, the hen y, and the chick z get the following equation:

a：5x+3y+1/3z = 100

b：x+y+z = 100

c：0 <= x <= 100

d：0 <= y <= 100

e：0 <= z <= 100

If solving this problem in the same way as an equation requires multiple guesses, one of the advantages of the computer is that the calculation speed is particularly violent and there is no regret, so we can bully her and ravage her! Therefore, we use the exhaustive method to solve the problem, which requires 101 3 guesses, but for the computer, a small case!

Question: A chicken is worth five, a hen is worth three, a chicken chick is worth one, a hundred dollars to buy a hundred chickens, ask the chicken, the mother, and the chick geometry.

Answer: 0 chickens, 25 hens, 75 chicks.

4 chickens, 18 hens, 78 chicks.

8 chickens, 11 hens, 81 chicks.

12 chickens, 4 hens, 84 chicks.

There is no special solution to this, only according to the traditional solution of the indefinite equation or the list method (make up the number) The solution of the indefinite equation on the network is as follows: The Problem of Hundred Chickens In the 6th century AD, there was a famous problem of 100 chickens in China's "Zhangqiu Jiansuanjing": "100 yuan to buy 100 chickens, 1 yuan for 3 chickens, 3 hens for 3 yuan, and 1 rooster for 5 yuan."

Q: How many chicks, hens, and roosters are there? ”

In the history of mathematics, this kind of problem is called the "Hundred Chicken Problem". How to solve the "100 chicken problem"? Suppose you buy x roosters, y hens, and z chicks, then according to the known conditions, there are:

x+y+z=100 (1)

5x+3y+z/3=100 (2)

In algebra, the number of equations in a system of multivariate linear equations is the same as the number of unknowns, and a system of equations generally has only one set of solutions. In this case, the system of equations contains fewer equations than the number of unknowns. A problem like this in which the number of equations is less than the number of unknowns is called an indefinite equation problem, and generally speaking, an indefinite equation has an infinite number of solutions.

So how does this system of equations be solved?

From (2) 3 (1):

14x+8y=200

That is, 7x+4y=100 (3).

In equation (3), 4y and 100 are both multiples of 4:

7x=100-4y=4(25-y)

Therefore 7x is also a multiple of 4, and 7 and 4 are coprimous, which means x must be a multiple of 4.

Let x 4t be substituted for (3) y=25-7t

Add x=4t again

y=25-7t

Substitution (1) has:

z=75+3t

Take t=1, t=2, t=3, and you have.

x=4 x=8 x=12

y=18 ｛ y=11 ｛ y=4

z=78 z=81 z=84

Because x, y, and z must all be less than 100 and all positive integers, only the above three sets of solutions fit the question. Although there are infinite sets of solutions for indefinite equations, there are only the above three sets of solutions in the "Hundred Chickens Problem".

0 Rooster 190 Hen 330 Chick 100 The meaning of the unequal sign is not necessarily all satisfied, according to the logical term The meaning is less than or equal to, and one of them can be satisfied

What he means is that those who meet the question must meet the following equation: 0 rooster 190, hen 330, chick 100, however, not all of them meet the above formula, which is a necessary but not sufficient condition.

Let the rooster be x, the hen y, and the chick z x+y+z=1005x+3y+z=100 to get x=0y=0z=100

The Baiji problem is an extremely famous mathematical problem in ancient China, and it is also one of the famous mathematical problems in the ancient world.

The question of 100 chickens comes from the ancient Chinese book "Zhang Qiu Jian Sutra", and the title is like this: 1 rooster is 5 yuan, 1 hen is 3 yuan, 3 chickens are 1 yuan, and 100 chickens can be bought for 100 yuan. Q: How many roosters, hens and chicks can I buy?

There are three answers.

4 roosters, 18 hens, 78 chicks;

8 roosters, 11 hens, 81 chicks;

12 roosters, 4 hens and 84 chicks.

Question: The chicken is worth five money, the hen is worth three, the chicken chick is worth one, and the chicken is worth 100 chickens.

Answer: 0 chickens, 25 hens, 75 chicks.

4 chickens, 18 hens, 78 chicks.

8 chickens, 11 hens, 81 chicks.

There are 12 chickens, 4 hens, and 84 chicks.

Today, there is a chicken Weng, which is worth a lot of money; The hen is one, and it is worth three; Three chickens are worth one. Where you buy 100 chickens for 100 dollars, ask the chickens, mothers, and catfishes? Answer:

Chicken four, worth twenty; Eighteen hens, worth fifty-four; Seventy-eight chickens are worth twenty-six. He also replied: Eight chickens, worth forty; The hen is eleven, worth thirty-three, and the chicken is eighty-one, worth twenty-seven.

And he replied: Twelve chickens, worth sixty; Hens.

Fourth, it is worth twelve; The chicken is eighty-four, and it is worth twenty-eight. ”

Generally speaking, an equation with more unknowns than the number of equations is an indefinite equation. In China's books such as "Sun Tzu's Arithmetic" and "Nine Chapters of Arithmetic", there are indefinite equation problems. The Hundred Chickens Problem in Zhang Qiu Jian's Arithmetic is a well-known indefinite equation problem for finding the solution of an integer number.

Zhang Qiujian lived during the Northern and Southern Dynasties in China. At an early age, he was good at thinking, intelligent and agile, and liked to solve mathematical problems, and was known as a "child prodigy". The prime minister at that time was very talented, so he thought of a "mystery of a hundred chickens" to investigate the level of the prodigy.

He called Zhang Qiujian's father and said, "Here is 100 yuan, buy me 100 chickens, and these 100 chickens should have roosters, hens and chicks." There can be no money left over and no more, and the number of chickens cannot be more or less.

At that time, a rooster was 5 cents, a hen was 3 cents, and three chicks were 1 cents. How can I buy 100 chickens with 100 dollars? Zhang Qiujian's father was a layman in arithmetic, and he told his son about it.

Xiao Qiu Jian thought about it and counted it on the ground. After a while, he told his father, "Just buy 4 roosters, 18 hens and 78 chicks."

Xiao Qiu Jian was summoned by the prime minister for his clever calculations, and he was rewarded. Zhang Qiujian studied more diligently, and finally became a famous mathematician, and compiled the "Zhang Qiujian Sutra of Calculations", which is one of the 10 important mathematical works of the Han and Tang dynasties in China.