This is Yang Hui Triangle, also called Jia Xian Triangle.
What he and we study most closely now is the coefficient law of binomial power expansion. As shown in the figure, in Jiaxian Triangle, the third number in the third row just corresponds to the square formula of the sum of two numbers (which will not be explained here).
Yang Hui Triangle is a triangular numerical table arranged by numbers, and its general form is as follows:
1 n=0
1 1 n= 1
1 2 1 n=2
1 3 3 1 n=3
1 4 6 4 1 n=4
1 5 10 10 5 1n = 5
1 6 15 20 15 6 1n = 6
…………………………………………………………
The most essential feature of Yang Hui Triangle is that its two hypotenuses are all composed of the number 1, and the other numbers are equal to the sum of the two numbers on its shoulders.
In fact, ancient mathematicians in China were far ahead in many important mathematical fields. The history of ancient mathematics in China once had its own glorious chapter, and the discovery of Yang Hui's triangle was a wonderful one.
Yang Hui was born in Hangzhou in the Northern Song Dynasty. In his book "Detailed Explanation of Algorithms in Nine Chapters" written by 126 1, he compiled a triangle table as shown above, which is called an "open root" diagram.
And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Specific purpose
A Brief History of Yang Hui Triangle: Jia Xian in the Northern Song Dynasty first used Jia Xian Triangle for high-order square root operation in about 1050, and Yang Hui, a mathematician in the Southern Song Dynasty, recorded and preserved Jia Xian Triangle in "Detailed Explanation of Nine Chapters" (196 1), so it was called Yang Hui Triangle. In Siyuan Jade Mirror (1303), Zhu Shijie, a mathematician in the Yuan Dynasty, developed the "A string triangle" into an "ancient seven-power diagram".
Time: Yang Hui (126 1 year) and Zhu Shijie (1303) can also clearly know that it was discovered by Yang Hui.
Zhu Shijie just expanded the content.
At the same time, this is also the law of quadratic coefficient of each term after polynomial (A+B) n is opened, that is
Therefore, the Y term of X layer of Yang Hui triangle is directly (y nCr x).
It is not difficult for us to get that the sum of all the terms in layer X is 2 (x- 1) (that is, when both A and B in (a+b) x are 1).
[the above y x refers to the x power of y; (a nCr b) refers to the number of combinations]
And such triangles are often used in our Olympic Games. The simplest thing is to ask you to find a way. Specific usage will be taught in the teaching content.
In foreign countries, this is also called Pascal Triangle.
S 1: These numbers are arranged in an isosceles triangle, and the numbers on both waists are 1.
S2: Oblique view from right to left, and oblique view from left to right, just like the previous view. I found that this series is symmetrical.
S3: The sum of the above two numbers is the number of the next line.
S4: Which row is this row, that is, the second number plus one. ……
The Rubik's Cube, also known as the vertical and horizontal map in China, has attracted countless people's fascination with it because of its magical characteristics. Judging from the legend of "river drawing, Luo writing and sage writing" in ancient China, the first person who systematically studied the Rubik's Cube was Yang Hui, an ancient algebra scientist in China.
Yang Hui, a native of Qiantang (now Hangzhou), was an outstanding mathematician in the Southern Song Dynasty. He and Qin, Zhu Shijie are also called the four great mathematicians in Song and Yuan Dynasties. He occupies a very important position in the history of ancient mathematics and mathematics education in China.
Yang Hui's research on Rubik's Cube originated from a short story. At that time, Yang Hui was a local official in Taizhou. On a voyage, he met a child who was in the way. When Yang Hui asked the reason, she realized that it was a child doing a math problem in the field. On hearing this, Yang Hui got off the sedan chair and asked the child what the problem was. It turns out that the child is solving an interesting problem given by an old gentleman: arrange the numbers from 1 to 9 in rows, and the result is equal to 15 regardless of vertical addition, horizontal addition and diagonal addition.
When Yang Hui saw this calculation, she remembered that she was also in the book Da Dai Li compiled by Dade, a scholar in the Western Han Dynasty.
Yes, I do. Yang Hui thought of this and came up with it with the children. It was not until the afternoon that they finally took out the formula.
Later, Yang Hui took the children to the old gentleman's house and talked with him about math problems. The old man said, "In the book Numerology Legacy written by Zhen Bende in the Northern Zhou Dynasty,' Nine palaces, two or four shoulders, six or eight feet, three left and seven right, wearing nine shoes, one and five in the middle. "'Yang Hui' listened, just like the children he released. He asked the old man, "Do you know how this Nine palace map did it?" The old man said he didn't know.
Tao.
Yang Hui returned home and pondered. One day, he finally found a rule, summed it up in four sentences: "Nine-child oblique, easy to go up and down, more harmonious left and right, four-dimensional prominent." That is to say, the nine numbers L ~ 9 are arranged obliquely in turn, and then the two numbers L and 9 are reversed, and the two numbers left 7 and right 3 are reversed. Finally, push out the four sides of 2, 4, 6 and 8, thus filling in the third-order Rubik's cube.
After Yang Hui worked out the construction method of the third-order magic square (also called Luo Shu or Nine palace map), he also systematically studied the fourth-order magic square to the tenth-order magic square. Among these magic squares, Yang Hui only gives the explanation of the construction methods of third-order and fourth-order magic squares. For the Rubik's Cube above the fourth order, Yang Hui only drew figures and did not leave the practice. But the fifth, sixth and even tenth order magic squares he drew are accurate, which shows that he has mastered the composition law of higher order magic squares.
Yang Hui Triangle also plays an important role in the information field.