The Shang Shu yun "poetry and poetry," poetry is the crystallization of human emotions. Some mathematicians are also poets of the heart, and many ancient mathematicians use poetry to express their nostalgia, compiling some of their favorite mathematical problems, as well as profound mathematical ideas and methods, into intriguing poetry problems. They use catchy poetry to enlighten the minds of future generations, transmit digital information, and make some abstract and difficult mathematical problems get vivid, vivid and rhyming poetic expressions, which are not only conducive to understanding, memorization, but also fascinating, stimulating the reader's strong interest and curiosity.
The ancient Chinese also liked to incorporate mathematical problems in ancient poems. "Algorithm Tongzong" is a popular and practical mathematical book, and it is also a representative work of incorporating numbers into poetry. This book was compiled by Cheng Dawei in the Ming Dynasty for nearly 20 years to compile a song and ballad recipe, and it is catchy to read. Cheng Dawei also has a drinking mathematical poem similar to the binary primary equation system: "Drinking in a row is chaotic, and the thin wine name is thick and mellow." A bottle of alcohol is drunk three times, and a thin bottle of wine is drunk. Together we drank nineteen, thirty-three drunken faces. I would like to ask a wise man, how much wine and how much alcohol?"
The poem is to say that a good bottle of wine can be drunk with 3 guests; Three bottles of thin wine to get drunk on one guest. If the 33 guests were drunk, they drank a total of 19 bottles of wine. I would like to ask: How many bottles of good wine and thin wine are there?
There is a problem in the famous "Sun Tzu Arithmetic Classic" that "things do not know their number". The original text of the arithmetic title is: "Now there is something that does not know its number, two of the three or three numbers, three of the five-five number, two of the seven-seven number, ask the geometry of the thing?" This problem has been passed down to later generations, and there have been many interesting names, such as "Ghost Valley Calculation" and "Han Xin Dianbing".
Example 1: A poem by the Ming Dynasty mathematician Wu Jing in the "Nine Chapters of Algorithmic Analogy" (1450):
Looking at the seventh floor of the tower, the red light doubled. A total of three hundred and eighty-one lights, please ask the tip of a few lights.
Example 2: A poem from the book Lilavolti (1150) by the ancient Indian mathematician Bhaskara:
The water of Pingping Lake is clear and can be seen, and the red lotus is half a foot on the surface. Out of the mud does not stain the pavilion, suddenly blown aside by the strong wind.
The fisherman watched busily forward, and the flowers were two feet away from the original position. Can be counted on the king please solve the problem, how to know the depth of the lake.
Example 3. Steak fish is counted
Three-inch fish nine mile ditch, mouth and tail to head. Ask how many fish there are, please say the reason to the king. ——Mei Zhencheng, "Adding and Deleting Algorithms to Unify Sects" (1761)
[Note] The size of the mile and step varies from dynasty to dynasty.
Zhou Dynasty 1 step = 8 feet. Qin Han to the North and South Dynasties 1 step = 6 feet, 1 mile = 300 steps. For example, the "Book of Han and Food Goods" contains: 1 mile = 300 steps, 1 step = 6 feet, referred to as the Qin and Han system. About from Sui and Tang dynasties to 1 mile = 360 steps, 1 step = 5 feet. This is the old system to create a ruler, five feet as a step. Mei Yancheng (1681-1763) Qing Dynasty mathematician, "瑴" sound tongduo, jué.
A group of small fish 3 inches long, they play in the river from mouth to tail, from beginning to end, lined up to 9 miles long. How many fish are there in this group? Please give the reason for your calculation.
[Solution] (ancient arithmetic solution) Because 1 mile = 360 steps, so 9 miles is
9× 360 = 3240 (step) and because 1 step = 5 feet = 50 inches
So 3240×50 = 162000 (inch) so 162000÷3 = 54000 (strip)
A: There are 54,000 of these lively and cute little fish.
[Description] Poetry, song, and mathematical poems are fascinating. This catchy mathematical poem seems to take us to the fun of watching swimming fish as children.
Example 4. Selling wine to hospitality (Xi Jiang Yue)
Hospitality with a pot of wine, do not know the pot inside the golden wave. Every time people are multiplied and harmonized, drink and fight together.
There are still five places to drink, and there is not much wine in the pot. To know the original wine is not bad pool, what method can be.
——Cheng Dawei,"Algorithm Tongzong"
[Comment] "Times" means 1 times. "Bucket half" refers to 1 bucket and a half, that is, 1.5 buckets.
Guests came to the house to buy wine, not knowing how much wine was left in the pot. I met an old friend at the place where I bought the wine, and after doubling the wine in the pot, I drank with my old friend, and each time I drank 1.5 buckets of wine in the pot. This is repeated 5 times, and finally the wine in the pot is exhausted. Ask how much wine is in the pot and how it is calculated.
[Solution] Ancient arithmetic writers used the "arithmetic method" to solve in ancient arithmetic books, and here they use a concise and clear "equation method" to solve.
The original pot has a wine x bucket, according to the inscription:
After "Doubling every person", after "drinking half of the bucket" 2x-1.5, "Drinking and drinking also through five places" after 2 (2 {2[2 (2x-1.5)-1.5] -1.5}-1.5) -1.5.
Finally the pot is exhausted, and the equation is obtained
2（2｛2［2（2x－1.5）－1.5］－1.5｝－1.5）－1.5＝0
Solution x = 1.453125 (bucket)
A: It turns out that there is wine in the pot 1 bucket 4 liters 5 in 3 spoons 1 copy 2 pinch 5 gui.
Example 5: Chickens and rabbits in the same cage
Now there are pheasant rabbits in the same cage, and there are thirty-five of them. There are ninety-four feet under it, ask the pheasant rabbit geometry?
——Sun Tzu Arithmetic, vol. 31
[Comment] "Pheasant", pronounced tongzhi, refers to pheasant, here refers to domestic chicken. Chickens and rabbits are kept in a cage, referred to as "chicken and rabbit in the same cage".
Today, there are chickens and rabbits in a cage, with 35 heads on the top and 94 chickens on the bottom.
[Solution] The solution of the "Sun Tzu Arithmetic Classic", after contemporary research, can be summarized as follows:
If the number of heads is 犃 and the number of feet (feet) is 犅, the solution formula is
兔数=1/2B-A 鸡数=A-(1/2B-A)
[Fun solution]: Let the rabbit and the chicken lift both feet at the same time, so that the feet in the cage will reduce the total number of heads× 2, because the chicken has only 2 feet, so only the two feet of the rabbit are left in the cage, and then ÷ 2 is the rabbit count. (Keyan)