Mathematical problems of lace of lantern mirror

The Mathematical Story in The Garden of Flowers in the Mirror

The Garden of Flowers in the Mirror is an encyclopedic classical novel written by Li Ruzhen in Qing Dynasty, which includes almost all the ancient books in A Collection of Classics and History, as well as knowledge of biology, horticulture, medicine, phonology, fortune telling, astronomy, geography, literature and mathematics. In the second half of the novel, there are 100 talented women, among whom Milanfen is a "divine operator". Let's discuss some math problems introduced in this novel.

First, divide the fruit.

In chapter 76 of the novel, it is said that many talented women visit Zongbo House. In the back garden, Dong Qingdian, Song Liangzhen, Stuart Wuer, Liao Xichun, Yan Yaochai and Jiang are playing abacus and discussing the algorithm. They worked out some arithmetic problems such as "Han Xin points pong" and "Twenty-eight nights for Kunyang". At this time, Liao Xichun gave everyone a question: "I saw you off at home, and my relatives and sisters came to see you off. Someone happened to send a plate of fresh fruit. Sisters are divided into seven or more sisters and eight or less sisters 16. How many sisters can count? How much fruit does each person share? "

Stuart Wuer replied: "This is a profit and loss algorithm, which is extremely easy: according to the first scheme, 1 fruit is added, and according to the second scheme, 16 fruit is lost. The difference between the two schemes is1+16 =17; At these two different points, everyone has 8-7= 1. So how many people will have the fruit of 17? The total difference between the two schemes divided by the difference between everyone in the two schemes is the number of people sharing the fruit. (1+16) ÷ (8-7) = 17 (person),17 is the number of people. According to the first distribution scheme:17× 7 =1kloc-0/9 (units),119+1=120 (units). Originally 17 people 120 fruit. "

The ancients summed up this solution as a song: "The operator wants to know the surplus, and the two companies take it from each other and succeed." The surplus is false, and the integral decreases each other. This is the law. The law divides things into prices, and the law divides people into real people. " In today's words, the formula of this song can be summarized as a formula: (abundance+deficit) ÷ the distribution difference between the two schemes = the number of participants. Then the number of distribution items is further calculated according to the number of participating distribution objects.

Students, do you understand?

Second, the brocade

In the 79th episode of the novel "Flowers in a Mirror", there is a story of "spreading brocade" and seeking a circle: several young ladies get together to talk about math. One of them, Qing Dian, pointed to the round table in front of him and asked, "Excuse me, sister, how many feet are there around this table?" The name of the person being asked is Milan Fen. She asked Baoyun around her for a ruler and measured the diameter of the round table, which was three feet two inches. Then he painted a "Lou Jin" (as shown in figure 1). After painting, he replied, "There are ten feet and four minutes around this table." (1 meter = 3 feet, 1 foot =? 10 foot, 1 foot = 10 inch, 1 inch = 10 minute).

1 on the left is the "carpet" drawn in the book "The Edge of Mirror Flowers", and Figure 2 on the right shows that I rewrote it into the current general vertical multiplication. As can be seen from the picture, "Dijin" is a big rectangle with some vertical and horizontal grid lines and diagonal lines connecting the diagonal lines of the grid, which is a bit like a carpet laid in a room, so it is vividly called "Dijin". By comparing the "ground floor" on the left and the vertical multiplication on the right, the actual content is almost the same. The vertical multiplicand and multiplier are written on the right and top of the large rectangular border in the "carpet" diagram respectively. Of the four sides of a large rectangle, the right and upper two are equivalent to the first horizontal line in vertical multiplication. Vertical, regardless of the decimal point for the time being, don't multiply the multiplicand 3 14 by the multiplier's 2 digits and 3 digits to get 628 and 942, and each multiplied number will occupy one line. The two rows are staggered to the right, and then aligned up and down. In the "carpet" diagram, two rows of vertical grids in a large rectangle are written from top to bottom, and each bit of the multiplicand is multiplied by each bit of the multiplicand to get 6, 2, 8 and 9, 3, 12. The product of these numbers and numbers, each number occupies one cell (the number with carry is written in the adjacent upper right cell). The numbers in these boxes are arranged vertically and horizontally. Add all the numbers on each diagonal and write the sum outside the rectangle.

In "Brocade", the three ones on the left vertical diagonal are actually "1" of multiplication or overtime carry. The last number of the vertical style is 10.048. In the "Dijin" map, it is at the lower left of the large rectangular frame, from upper left to lower right (that is, counterclockwise from upper left), and read it together. The left and bottom of a large rectangle are equivalent to the second vertical horizontal line. After drawing a "carpet" map, it is equivalent to writing a vertical line of multiplication.

Therefore, Milanfen in The Mirror Flower Edge can see that the circumference of the round table is 10.004 minutes and 8 centimeters (≈3.35 meters) after painting the carpet.

Third, calculate the cup weight.

In the back garden of Zongbofu, Baoyun pointed to a set of gold cups on the table and said to the talented women, "This cup is size nine. I used 126 gold to ask craftsmen to build them, and the weight of these cups increased exponentially from small to large. Sister, can you work out the size and weight of the cup? "

Lanfen said: "This is the difference method. The method is to add 9 to 1 do 10 and multiply 9 by 10 to make 90 copies, making 45 copies. Use 126÷45=2.8 (two), which is 228 yuan (1 two = 10 yuan). This ninth small cup is so heavy. " Immediately, I took out two counting books from the small counting bag brought by the maid and wrote them with a pen. The large cup weighs 25.22 yuan, the two cups weigh 22.24 yuan, the three cups weigh 19.26 yuan, the four cups weigh 16.28 yuan, the five cups weigh 14 yuan, the six cups weigh1.22 yuan, and the seven cups weigh 820.

Baoyun was surprised and asked, "Sister, how can you see the weight of each cup at a glance?"

Students, actually Milanfen thinks this way: The weight of these nine cups is increasing exponentially, so their weight happens to be a arithmetic progression. She looked at the weight of the small cup as 1, the other eight cups as 2, 3, 4 ... 8, and the total number of 9.9 cups as 1+2+3+4+...+8+9 = (1+9) × 9 ÷/kloc-

Baoyun ordered people to weigh him, and it was really good.

Fourth, count lanterns.

After calculating the circumference of the table and the weight of the golden cup, the talented women, led by the hostess Bian Baoyun, came to Xiao 'ao Mountain in the back garden to watch the colorful lights. I saw twenty-seven buildings connected in series on three sides, with low corridors in the south and light balls hanging upstairs and downstairs. All kinds of patterns, bright colors and brilliant brilliance, like stars, rise and fall, one after another, which is overwhelming.

Bian Baoyun said: "There are two kinds of lights upstairs: one has three big balls above, six small balls below, and nine big and small balls are Yi Deng; The other has three big balls on the top and 18 small balls on the bottom, and the size of 2 1 is1; Big ball 396, small ball 1440. There are also two kinds of lights downstairs: one is that there are 1 big balls above and two small balls below; The other is that there are 1 big balls on it and 4 small balls on the bottom. Big ball 360, small ball 1200. Do you know how many different types of ball lights are upstairs and downstairs? "

A talented woman said, "We know how many lamps there are by looking and counting."

Milan Fen said: "I think we can solve it by solving the problem of' chickens and rabbits in the same cage'. Take the lights downstairs as an example: the number of small light bulbs 1200 is halved to get 600, the number of lights decorated with four small light bulbs minus 360 is 240, and the number of lights decorated with two small light bulbs minus 240 is 120.

Then, everyone asked Milan Fen to calculate the number of two kinds of lights upstairs. Bian Baoyun asked the servant to take out the light list and found that everyone's calculations were not bad at all.

In fact, the "chicken and rabbit problem" originated from Sun Tzu's calculation. It is said that when he was young, his grandson, who wrote the classic Art of War, visited a friend's house and saw that there were many chickens and rabbits in his friend's house. He casually asked, "How many chickens and rabbits do you have at home?" The friend replied, "There are thirty-five chickens and rabbits, and the number of feet is ninety-four. Please count the chickens and rabbits. " Grandson was very interested, and after several thoughts, he finally found the answer.

Sun Tzu's solution is to assume that the chicken is "independent of the golden rooster" (one foot touches the ground) and the rabbit is "Yue Bai" (the front foot is raised and the rear two feet touch the ground), that is, the chicken and the rabbit each cut off half of their feet, that is, their total feet are only 94÷2=47 (only). At this time, the number of chickens is equal to the number of feet, but the number of each rabbit and foot is 60. So now the difference between the total number of heads and the total number of feet is 47-35= 12 (only). Obviously, this 12 is the number of rabbits. So the chicken is 35- 12=23 (only)

Is your Milan solution reasonable? Can you figure out how many lamps there are in each of the two kinds of lamps upstairs? If you can work it out correctly, you must be a gifted scholar or talented woman.