There are five * * * wells today, but one or two wells are not enough, such as those in Otsuichi; B three won't be enough, like C one won; C-4 is insufficient, such as d-1; Ding and other five are not enough, and one wins; Missing five or six is like missing one. If everyone is short of an award, they will all be caught. Find the geometry of deep well length!
Of course, it takes rope to get water from the well with a bucket. "Bi" means "rope". The original question means:
Five families use a well. The depth of the well is 1 rope length, which is longer than the length of two rope lengths. It is longer than the three ropes of group B 1 rope length; 1 dingjia rope is longer than 4 C rope; Longer than five ropes of Dingjia 1 rope length; It is 1 rope length, which is longer than 6 ropes. If each family adds another water rope, it will be just right to take water. What is the depth of the well and the length of the water intake rope?
Although the problem is fictitious, it is the earliest indefinite equation problem.
With modern symbols, the rope lengths of A, B, C, D and E can be set as X, Y, Z, U and V respectively; And deep H. according to the meaning of the question, you can get
2x+y=h，
3y+z=h，
4z+u=h，
5u+v=h，
6v+x=h .
This is a set of equations with 6 unknowns and 5 equations. An equation (or equation) with more unknowns than equations is called an indefinite equation. It can be obtained by addition, subtraction and elimination.
x=26572 1h，y= 19 172 1h，z= 14872 1h，
u= 12972 1h，v=7672 1h .
Given different values of H, we can get different values of X, Y, Z, U, V, and we can get the group solution as long as some specific conditions are given. Only one set of solutions is given in the original book, which is the smallest positive integer solution.
Ancient mathematicians in China made brilliant achievements in indefinite equations on the basis of nine chapters of arithmetic. The problem of "five * * * wells" is the predecessor of Baiji technology, and it is also a great development technology.
The problem of "five * * * wells" has attracted the attention of many mathematicians in the world. In the western history books of mathematics, the earliest contribution to the study of indefinite equations is attributed to Diophantine in Greece. In fact, he didn't start to study these problems until around 250 AD, which was more than 200 years later than our country.
In the first half of the 6th century AD, Zhang Qiujian had the problem of one hundred chickens in his Calculation of Zhang Qiujian: today there is a chicken worth five; One mother hen is worth three; When a chicken is born, it is worth one. For every hundred dollars, buy a hundred chickens. What is the geometric relationship between rooster, mother and chicken?
It means that if 1 cock is worth 5 yuan; 1 hen is worth 3 yuan; The value of three chickens 1 yuan. I bought 100 chickens for 100 yuan. How much is a rooster, a hen and a chicken?
Let the cock, hen and chicken be x, y and z respectively, then it is not difficult to get the indefinite equation and eliminate z.
5x+3y+ 13z= 100
x+y+z= 100
Eliminating z is not difficult to get.
y=7x4
Because y is a positive integer, x must be a multiple of 4.
Let x = 4t, then y = 25-7t and z = 75+3t.
∵x>0,∴4t>0,t>0；
∵ y > 0，∴ 25-7t > 0，t < 347。
So t = 1, 2, 3.
∴ The original equation has three sets of answers:
{x=4，y= 18，z=78 {x=8，y= 1 1，z=8 1 {x= 12，y=4，z=84
Mathematical historians have commented that there are multiple sets of answers to an application problem, which is unprecedented in the history of mathematics, and the problem of "hundred chickens" is a precedent. Zhang Qiujian's Su 'an Sutra did not give a solution, only said: "Shu Yue: every rooster increases by four, every hen decreases by seven, and every chicken benefits by three, and you get it." It means: if you buy seven hens less, you can buy four cocks and three chickens more. Because seven hens are worth 2 1 and four cocks are worth 20, which is the difference between three chickens. As long as you get one set of answers, you can deduce the other two sets. But how did this solution come about? There is no explanation in the book. Therefore, the so-called "hundred chickens technique", that is, the solution to the problem of hundred chickens, has aroused great interest.
Later, Zhen Luan put forward two "Hundred Chicken Problems" in his book Numerology Legacy, the meaning of which is the same as the original one, but the numbers are different. Yang Hui, a famous mathematician in the Song Dynasty, also cited a similar question in his book Algorithm for Extracting Odds from Ancient Times:
"One hundred dollars of warm orange, green orange and flat orange * * * one hundred pieces. Only Yunnuan Orange has seven articles, Green Orange has three articles and Pingorange has three articles. Ask for geometry? "
During the Ming and Qing dynasties, some people put forward the "hundred chickens problem" of more than three yuan. However, all the books, such as Zhang Qiu's "Concurrent Calculations", do not give a general method to solve the problem.
In the 7th century, someone proposed another way to solve the problem, but it was only a matter of numbers. In Qing Dynasty, Jiao Xun pointed out his mistakes in the book Interpretation of Addition, subtraction, multiplication and division. Since then, new solutions have been proposed, but none of them have been completely universally solved. For example, Ding Qu-zhong gave a simple solution in his Addendum to Mathematics: Assuming there is no rooster, buy 100 hens and chickens with the money of 100, and get 25 hens and 75 chickens. Now buy seven hens less and four cocks and three chickens more, and you will get the first set of answers. Similarly, two other groups can also be introduced. It was not until the19th century that people combined this kind of problem with the study of "developing greatly and seeking a skill".
Hundred chickens is a famous historical topic, which has great influence in the world. Similar topics are also common abroad.