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中國哲學書電子化計劃
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《卷下》

 英文翻譯：人工智能和中國哲學書電子化計劃用戶 [?] 電子圖書館
1 卷下:
今有甲、乙、丙、丁、戊、己、庚、辛、壬九家共翰租。甲出三十五斛，乙出四十六斛，丙出五十七斛，丁出六十八斛，戊出七十九斛，己出八十斛，庚出一百斛，辛出二百一十斛，壬出三百二十五斛。凡九家共翰租一千斛。僦運直折二百斛外，問家各幾何？
Now there are Jia, Yi, Bing, Ding, Wu, Ji, Geng, Xin, and Ren nine families jointly paying rent. Jia contributed 35 hu, Yi contributed 46 hu, Bing contributed 57 hu, Ding contributed 68 hu, Wu contributed 79 hu, Ji contributed 80 hu, Geng contributed 100 hu, Xin contributed 210 hu, and Ren contributed 325 hu. In total, the nine families jointly paid a rent of 1,000 hu. After deducting the cost of hiring transport, which amounted to 200 hu, how much should each family contribute?
答曰：甲二十八斛。乙三十六斛八蚪。丙四十五斛六蚪。丁五十四斛四蚪。戊六十三斛二蚪。己六十四斛。庚八十斛。辛一百六十八斛。壬二百六十斛。
Answer: Jia contributed 28 hu. Yi contributed 36 hu and 8 dou. Bing contributed 45 hu and 6 dou. Ding contributed 54 hu and 4 dou. Wu contributed 63 hu and 2 dou. Ji contributed 64 hu. Geng contributed 80 hu. Xin contributed 168 hu. Ren contributed 260 hu.
術曰：置甲出三十五斛，以四乘之，得一百四十斛。以五除之，得二十八斛。乙出四十六斛，以四乘之，得一百八十四斛。以五除之，得三十六斛八蚪。丙出五十七斛，以四乘之，得二百二十八斛。以五除之，得四十五斛六蚪。丁出六十八斛，以四乘之，得二百七十二斛。以五除之，得五十四斛四斟。戊出七十九斛，以四乘之，得三百一十六斛。以五除之，得六十三斛二斟。己出八十斛，以四乘之，得三百二十斛。以五除之，得六十四斛。庚出一百斛，以四乘之，得四百斛。以五除之，得八十斛。辛出二百一十斛，以四乘之，得八百四十斛。以五除之，得一百六十八斛。壬出三百二十五斛，以四乘之，得一千三百斛。以五除之，得二百六十斛。
Method: Take Jia's contribution of 35 hu and multiply it by four, resulting in 140 hu. Divide this by five, yielding 28 hu. Yi's contribution of 46 hu is multiplied by four, resulting in 184 hu. Divide this by five, obtaining 36 hu and 8 dǒu. Bing's contribution of 57 hu is multiplied by four, yielding 228 hu. Divide this by five, resulting in 45 hu and 6 dǒu. Ding's contribution of 68 hu is multiplied by four, giving 272 hu. Divide this by five, getting 54 hu and four zhen. Wu's contribution of 79 hu is multiplied by four, obtaining 316 hu. Divide this by five, giving 63 hu and two zhen. Ji's contribution of 80 hu is multiplied by four, getting 320 hu. Divide this by five, which results in 64 hu. Geng's contribution of one hundred hu is multiplied by four, producing four hundred hu. Divide this by five to get eighty hu. Xin's contribution of 210 hu is multiplied by four, amounting to 840 hu. Divide this by five and obtain 168 hu. Ren's contribution of 325 hu is multiplied by four, totaling one thousand three hundred hu. Divide this by five and get two hundred sixty hu.

2 卷下:
今有丁一千五百萬，出兵四十萬。問幾丁科一兵？
Now there are 15 million households, contributing 400,000 soldiers. The question is: how many households contribute one soldier?
答曰：三十七丁五分。
Answer: 37.5 households.
術曰：置丁一千五百萬，為實。以兵四十萬為法。實如法，即得。
Method: Take the number of households, one thousand five hundred million, as the dividend. Use the 400,00 soldiers as the divisor. Divide the dividend by the divisor, and the result is obtained directly.

3 卷下:
今有平地聚粟，下周三丈六尺，高四尺五寸。問粟幾何？
Now there is a pile of millet on flat ground, with a base circumference of three zhang six chi and a height of four chi five cun. How much millet is there?
答曰：一百斛。
Answer: 100 hu.
術曰：置周三丈六尺，自相乘，得一千二百九十六尺。以高四尺五寸乘之，得五千八百三十二尺。以三十六除之，得一百六十二尺。以斛法一尺六寸二分除之，即得。
Method: Take the base circumference of three zhang and six chi, multiply it by itself, resulting in 1296 square chi. Multiply this by the height of four chi and five cun, obtaining 5832 cubic chi. Divide this by thirty-six, resulting in one hundred sixty-two cubic chi. Divide this by the hu standard of 1 chi six cun two fen to obtain the result directly.

4 卷下:
今有佛書凡二十九章，章六十三字。問字幾何？
Now there is a Buddhist scripture with a total of twenty-nine chapters, each chapter containing sixty-three characters. How many characters are there in total?
答曰：一千八百二十七。
Answer: 1,827.
術曰：置二十九章，以六十三字乘之，即得。
Method: Take the twenty-nine chapters and multiply them by sixty-three characters per chapter; this gives the total directly.

5 卷下:
今有諪局方一十九道。問用諪幾何？
Now there is a chessboard with a square pattern of nineteen lines on each side. How many pieces are used?
答曰：三百六十一。
Answer: 361.
術曰：置一十九道，自相乘之，即得。
Method: Take the nineteen lines, multiply them by themselves, and you get the result directly.

6 卷下:
今有租九萬八千七百六十二斛，欲以一車載五十斛，問用車幾何？
Now there is a total of 98,762 hu of rent, and it is desired to transport one cartload of fifty hu per vehicle. How many carts are needed?
答曰：一千九百七十五乘，奇一十二斛。
Answer: 1975 carts will be used, with a remainder of twelve hu.
術曰：置租九萬八千七百六十二斛，為實。以一車所載五十斛為法。實如法，即得。
Method: Take the total rent of ninety-eight thousand seven hundred and sixty-two hu as the dividend. Use fifty hu, the amount one cart can carry, as the divisor. Divide the dividend by the divisor to obtain the result directly.

7 卷下:
今有丁九萬八千七百六十六，凡二十五丁出一兵。問兵幾何？
Now there are 98,766 households, and for every twenty-five households, one soldier is contributed. How many soldiers are there?
答曰：三千九百五十人，奇一十六丁。
Answer: 3,950 soldiers, with a remainder of sixteen households.
術曰：置丁九萬八千七百六十六，為實。以二十五為法。實如法，即得。
Method: Take the number of households as ninety-eight thousand seven hundred sixty-six and set it as the dividend. Use twenty-five as the divisor. Divide the dividend by the divisor and you will obtain the result directly.

8 卷下:
今有絹七萬八千七百三十二匹，令一百六十二人分之。問人得幾何？
Now there are 78,732 bolts of silk to be divided among 162 people. How many bolts does each person get?
答曰：四百八十六匹。
Answer: 486 bolts.
術曰：置絹七萬八千七百三十二匹，為實。以一百六十二人為法。實如法，即得。
Method: Take the silk of seventy-eight thousand seven hundred thirty-two bolts as the dividend. Use 162 people as the divisor. Divide the dividend by the divisor directly to obtain the result.

9 卷下:
今有三萬六千四百五十四戶，戶輸綿二斤八兩。問計幾何？
Now there are 36,454 households, each household is required to contribute two jin and eight liang of cotton. What is the total amount?
答曰：九萬一千一百三十五斤。
Answer: 91,135 jin.
術曰：置三萬六千四百五十四戶，上十之，得三十六萬四千五百四十。以四乘之，得一百四十五萬八千一百六十兩。以十六除之，即得。
Method: Take the number of households (36,454), multiply it by ten, resulting in 364,540. Multiply this by four to get 1,458,160 liang. Divide this by sixteen, and the result is directly obtained.

10 卷下:
今有綿九萬一千一百三十五斤，給與三萬六千四百五十四戶。問戶得幾何？
Now there are 91,135 catties of cotton to be distributed among 36,454 families. How many catties does each family receive?
答曰：二斤八兩。
Answer: 2 jin and 8 liang.
術曰：置九萬一千一百三十五斤，為實。以三萬六千四百五十四戶為法。除之，得二斤，不盡一萬八千二百二十七斤。以一十六乘之，得二十九萬一千六百三十二兩。以戶除之，即得。
Method: Take the 91,135 pounds of cotton as the dividend. Use the number of families, 36,454, as the divisor. Divide it and get two jin, with a remainder of eighteen thousand two hundred and twenty-seven jin. Multiply the remainder by sixteen to obtain 291,632 liang. Divide this amount by the number of families and you will get the result directly.

11 卷下:
今有粟三千九百九十九斛九蚪六升，凡栗九蚪易豆一斛。問計豆幾何？
Now there are 3,999 hu nine dǒu six sheng of millet. For every nine dǒu of millet, it can be exchanged for one hu of broad beans. How many hu of broad beans are there in total?
答曰：四千四百四十四斛四斟。
Answer: 4,444 hu and four zhen (of broad beans).
術曰：置粟三千九百九十九斛九斗六升，為實。以九斗為法。實如法。即得。
Method: Take the millet of three thousand nine hundred ninety-nine hu, nine dǒu and six sheng as the dividend. Use nine dǒu as the divisor. Divide the dividend by the divisor. The result is obtained directly.

12 卷下:
今有粟二千三百七十四斛，斛加三升。問共粟幾何？
Now there are 2,374 hu of millet, with an additional three sheng per hu. What is the total amount of millet?
答曰：二千四百四十五斛二斗二升。
Answer: 2,445 hu two dǒu and two sheng.
術曰：置粟二千三百七十四斛，以一斛三升乘之，即得。
Method: Take the millet amount of 2,374 hu, multiply it by one hu and three sheng, and you will directly obtain the result.

13 卷下:
今有粟三十六萬九千九百八十斛七斗，在倉九年，年斛耗三升。問一年、九年各耗幾何？
Now there are 369,980 hu seven dǒu of millet in the granary, stored for nine years, with an annual loss of three sheng per hu. Now there are 369,980 hu and seven dǒu of millet stored in the granary for nine years, with an annual loss of three sheng per hu. How much is lost in one year, and how much in nine years?
答曰：一年耗一萬一千九十九斛四斗二升一合。九年耗九萬九千八百九十四斛七斗八升九合。
Answer: In one year, the loss amounts to 11,994 hu four dǒu two sheng and one ge. In nine years, the total loss is 99,894 hu seven dǒu eight sheng and nine ge.
術曰：置三十六萬九千九百八十斛七斗，以三升乘之得一年之耗。又以九乘之，即九年之耗。
Method: Take 369,987 hu seven dǒu, multiply it by three sheng to obtain the loss for one year. Multiply this result by nine again, and you will get the total loss over nine years.

14 卷下:
今有貸與人絲五十七斤，限歲出息一十六斤。問斤息幾何？
Now there is a loan of 57 jin of silk to someone, with an interest requirement of 16 jin per year. What is the interest rate per jin?
答曰：四兩五十七分兩之二十八。
Answer: Four liang and 28/57 of a liang.
術曰：列限息絲一十六斤，以一十六兩乘之，得二百五十六兩。以貸絲五十七斤除之⑤。不盡，約之，即得。
Method: Take the interest silk of sixteen jin, multiply it by sixteen liang to get 256 liang. Divide this by the loaned silk of fifty-seven jin. If it does not divide evenly, approximate it; thus you obtain the result.

15 卷下:
今有三人共車，二車空；二人共車，九人步。問人與車各幾何？
Now there are three people sharing a cart, and two carts remain empty; two people share another cart, while nine people walk. What are the numbers of people and carts?
答曰：一十五車。三十九人。
Answer: Fifteen carts. Thirty-nine people.
術曰：置二車，以三乘之，得六。加步者九人，得車一十五。欲知人者，以二乘車，加九人，即得。
Method: Take two carts, multiply by three to get six. Add the nine people walking, resulting in fifteen carts. To find out the number of people, multiply two by the number of carts and add nine people; thus you get the result.

16 卷下:
今有粟一十二萬八千九百四十斛九斗三合，出與人買絹，一匹直粟三斛五斗七升。問絹幾何？
Now there are 128,940 hu nine dǒu three ge of millet given to someone for buying silk, with one bolt of silk valued at three hu five dǒu seven sheng of millet. What is the amount of silk?
答曰：三萬六千一百一十七匹三丈六尺。
Answer: 36,117 bolts three zhang six chi.
術曰：置粟一十二萬八千九百四十斛九斗三合，為實。以三斛五斗七升為法。除之，得匹。餘四十之，所得，又以法除之，即得。
Method: Take the millet of 128,949 hu nine dǒu and three ge as the dividend. łoż Use three hu five dǒu and seven sheng as the divisor. Divide it, and you get the number of bolts. Take the remainder, multiply it by forty, then divide again using the same divisor; thus you obtain the final result.

17 卷下:
今有婦人河上蕩桮。津吏問曰：「桮何以多？」婦人曰：「家有客。」津吏曰：「客幾何？」婦人曰：「二人共飯，三人共羹，四人共肉，凡用聣六十五。不知客幾何？」
Now there is a woman washing cups at the riverbank. The ferry official asked, "Why are there so many cups?" The woman replied, "There are guests at my home." The ferry official asked, "How many guests?" The woman said: "Two people share a meal, three people share soup, four people share meat, and in total sixty-five cups are used. I do not know how many guests there are."
答曰：六十人。
Answer: Sixty people.
術曰：置六十五桮，以一十二乘之，得七百八十，以十三除之，即得。
Method: Take the 65 cups, multiply by twelve to get 780, then divide by thirteen; thus you obtain the answer.

18 卷下:
今有木，不知長短。引繩度之，餘繩四尺五寸。屈繩量之，不足一尺。問木長幾何？
Now there is a piece of wood whose length is unknown. Using a rope to measure it, the rope has 4 chi and 5 cun left over. Folding the rope to measure it results in one chi short. What is the length of the wood?
答曰：六尺五寸。
Answer: Six chi and five cun.
術曰：置餘繩四尺五寸，加不足一尺，共五尺五寸。倍之，得一丈一尺。減餘四尺五寸，即得。
Method: Take the remaining rope of four chi and five cun, add to it the one chi that was insufficient, making a total of five chi and five cun. Double this amount, resulting in eleven chi. Subtract the remaining four chi and five cun; thus you obtain the length of the wood.

19 卷下:
今有器中米，不知其數。前人取半，中人三分取一，後人四分取一，餘米一斗五升。問本米幾何？
Now there is rice in a vessel, but its quantity is unknown. The first person took half, the second person took one-third of what remained, the third person took one-fourth of what was left after that, and there were 1 dǒu five sheng of rice remaining. What was the original amount of rice?
答曰：六斗。
Answer: Six dǒu.
術曰：置餘米一斗五升，以六乘之，得九斗。以二除之，得四斗五升。以四乘之，得一斛八斗。以三除之，即得。
Method: Take the remaining rice of one dǒu five sheng, multiply it by six to get nine dǒu. Divide this amount by two, resulting in four dǒu and five sheng. Multiply this result by four to get one hu eight dǒu. Divide this amount by three; thus you obtain the original quantity of rice.

20 卷下:
今有黃金一斤，直錢一十萬。問兩直幾何？
Now there is one jin of gold, valued at 100,000 qian. What is the value per liang?
答曰：六千二百五十錢。
Answer: 6,250 qian.
術曰：置錢一十萬，以一十六兩除之，即得。
Method: Take the money amount of one hundred thousand qian and divide it by sixteen liang; thus you obtain the value per liang.

21 卷下:
今有錦一疋，直錢一萬八千。問丈、尺、寸各直幾何？
Now there is one bolt of brocade, valued at 18,000 qian. Now there is one bolt of brocade, valued at 18,000 qian. What are the values per zhang, chi, and cun?
答曰：丈，四千五百錢。尺，四百五十錢。寸，四十五錢。
Answer: One zhang is worth 4,500 qian. One chi is worth 450 qian. One cun is worth forty-five qian.
術曰：置錢一萬八千，以四除之得尺、寸之直。得一丈之直。一退、再退，得尺、寸之直。
Method: Take the money amount of 18,000 and divide it by four to obtain the value per chi and cun. This gives you the value for one zhang. One step back, then another step back, to obtain the values per chi and cun.

22 卷下:
今有地長一千步，廣五百步。尺有鶉，寸有鷃。問鶉、鷃各幾何？
Now there is a field that is 1,000 bu long and 500 bu wide. There are quail per chi and yàn birds per cun. What are the numbers of quail and yàn?
答曰：鶉一千八百萬。鷃一億八千萬。
Answer: 18 million quail. Yàn birds, one hundred and eighty million.
術曰：置長一千步，以廣五百步乘之，得五十萬步。以三十六乘之，得一千八百萬尺，即得鶉數。上十之，即得赉數。
Method: Take the length of 1,000 bu, multiply it by the width of 500 bu to get fifty thousand bu. Multiply this by thirty-six to get eighteen million chi, which is the number of quail. Increase it tenfold, and you obtain the number of yàn birds.

23 卷下:
今有六萬口，上口三萬人，日食九升；中口二萬人，日食七升；下口一萬人，日食五升。問上、中、下口共食幾何？
Now there are 60,000 people, with 30,000 being adults who consume nine sheng of food per day; 20,000 are middle-aged individuals who consume seven sheng per day; 10,000 are children who consume five sheng per day. What is the total daily food consumption for adults, middle-aged individuals, and children combined?
答曰：四千六百斛。
Answer: 4,600 hu.
術曰：各置口數，以日食之數乘之。所得，并之，即得。
Method: Take the number of each category of people and multiply it by their daily food consumption. Add up all the results, and you will obtain the total amount.

24 卷下:
今有方物一束，外周一匝有三十二枚。問積幾何？
Now there is a bundle of square-shaped items; one full layer around the outside contains thirty-two pieces. What is the total number?
答曰：八十一枚。
Answer: Eighty-one pieces.
術曰：重置二位。上位減八，餘加下位。至盡虛加一，即得。
Method: Set up two positions again. Subtract eight from the upper position, and add the remainder to the lower position. Continue until it is exhausted; then add one more at the end, and you will obtain the result.

25 卷下:
今有竿不知長短，度其影得一丈五尺。別立一表，長一尺五寸，影得五寸。問竿長幾何？
Now there is a pole of unknown length; measuring its shadow gives 1 zhang five chi. Set up another gnomon one chi and five cun tall, whose shadow measures five cun. What is the length of the pole?
答曰：四丈五尺。
Answer: Four zhang and five chi.
術曰：置竿影一丈五尺，以表長一尺五寸乘之，上十之，得二十二丈五尺。以表影五寸除之，即得。
Method: Take the pole's shadow of one zhang and five chi, multiply it by the gnomon height of one chi and five cun, then increase this result tenfold to get twenty-two zhang and five chi. Divide this amount by the gnomon's shadow of five cun; thus you obtain its length.

26 卷下:
今有物，不知其數。三、三數之，賸二；五、五數之，賸三；七、七數之，賸二。問物幾何？
Now there is an object, but we do not know its quantity. Counting it in groups of three leaves a remainder of two; counting it in groups of five leaves a remainder of three; Counting it in groups of seven leaves a remainder of two. What is the quantity of the object?
答曰：二十三。
Answer: Twenty-three.
術曰：「三、三數之，賸二」，置一百四十；「五、五數之，賸三」，置六十三；「七、七數之，賸二」，置三十。并之，得二百三十三。以二百一十減之，即得。凡三、三數之，賸一，則置七十；五，五數之，賸一，則置二十一；七、七數之，賸一，則置十五。一百六以上，以一百五減之，即得。
Method: "Counting in groups of three, remainder two," set down one hundred and forty; "Counting in groups of five, remainder three," set down sixty-three; "Counting in groups of seven, remainder two," set down thirty. Add them together to get two hundred and thirty-three. Subtract two hundred and ten from it, and you will obtain the answer. Generally, if counting in groups of three leaves remainder one, set down seventy; if counting in groups of five leaves remainder one, set down twenty-one; if counting in groups of seven leaves remainder one, then set down fifteen. If the total is more than 160, subtract 150 from it; thus you obtain the answer.

27 卷下:
今有獸六首四足，禽四首二足。上有七十六首，下有四十六足。問篱、獸各幾何？
Now there are beasts with six heads and four feet, and birds with four heads and two feet. Above there are seventy-six heads, below there are forty-six feet. Ask: How many birds and beasts are there?
答曰：八獸，七禽。
Answer: Eight beasts, seven birds.
術曰：倍足以減首，餘，半之之，即禽。即獸。以四乘獸，減足，餘，半
Method: Double the number of feet and subtract it from the number of heads; half the remainder is the number of birds. The rest are beasts. Multiply the number of beasts by four, subtract it from the total feet; half

28 卷下:
今有甲、乙二人，持錢各不知數。甲得乙中半，可滿四十八。乙得甲太半，亦滿四十八。問甲、乙二人元持錢各幾何？
Now there are two people, Jia and Yi, each holding an unknown amount of money. If Jia receives half of what Yi has, he will have a total of forty-eight. If Yi receives two-thirds of what Jia has, he too will have a total of fortyeight. Ask: Originally, how much money did Jia and Yi each hold?
答曰：甲持錢三十六。乙持錢二十四。
Answer: Jia originally held thirty-six units of money. Yi originally held twenty-four units of money.
術曰：如方程求之。置二甲、一乙、錢九十六，於右方。置二甲、三乙、錢一百四十四，於左方。以右方二乘左方，上得四，中得六，下得二百八十八錢。以右行再減左行，左上空，中餘四乙，為法；下餘九十六錢，為實。上法，下實，得二十四錢，為乙錢。以減右下九十六，餘七十二，為實；以右上二甲為法。上法、下實，得三十六，為甲錢也。
Method: Solve it using the method of equations. Place two Jia, one Yi, and ninety-six units of money on the right side. Place two Jia, three Yi, and one hundred forty-four units of money on the left side. Multiply the right side by two times the left side, resulting in four at the top, six in the middle, and 288 units of money at the bottom. Subtract twice the right row from the left row; the top on the left is empty, and the middle has four Yi remaining as the divisor; the bottom has ninety-six units of money remaining as the dividend. Apply the divisor to the dividend, resulting in twenty-four units of money, which is Yi's amount. Subtract this from ninety-six at the bottom on the right; the remainder is seventy-two as the dividend; take two Jia at the top of the right side as the divisor. Apply the divisor to the dividend and obtain thirty-six, which is Jia's amount.

29 卷下:
今有百鹿入城，家取一鹿，不盡；又三家共一鹿，適盡。問城中家幾何？
Now there are one hundred deer entering a city; each household takes one deer, but they do not finish; then three households share one deer and it is just finished. Ask: How many households are in the city?
答曰：七十五家。
Answer: Seventy-five households.
術曰：以盈不足取之。假令七十二家，鹿不盡四。令之九十家，鹿不足二十。置七十二於右上，盈四於右下。置九十於左上，不足二十於左下。為維乘之凹，所得，并為實。并盈不足為法。除之，即得。
Method: Solve it using the principle of surplus and deficit. Assume there are seventy-two households, then four deer remain unclaimed. If we assume ninety households, twenty fewer deer would be needed. Place seventy-two at the upper right and four surplus at the lower right. Place ninety at the upper left and twenty deficit at the lower left. Multiply crosswise, add the results to obtain the dividend. Add the surplus and deficit together as the divisor. Divide, and you obtain the answer.

30 卷下:
今有三雞共啄粟一千一粒。雛啄一，母啄粟。三雞主各償幾何？
Now there are three chickens pecking at one thousand and one grains of millet. A chick pecks one grain, a hen pecks ten grains. How many grains should each of the three chickens compensate?
答曰：雞雛主一百四十三。雞母主二百八十六。雞翁主五百七十二。翁啄四。主責本
Answer: The chick's owner compensates 143. The hen's owner compensates 286. The rooster's owner compensates 572. A rooster pecks four grains. The owner is responsible for the original amount.
術曰：置粟一千一粒，為實。副并三雞所啄粟七粒，為法。除之，得一百四十三粒，為雞雛主所償之數。遞倍之，即得母、翁主所償之數。
Method: Place one thousand and one grains as the dividend. Add up the total number of grains pecked by three chickens, which is seven grains, as the divisor. Divide it to get 143 grains, which is the amount that the chick's owner should compensate. Multiply this by two successively, and you will obtain the amounts that the hen and rooster owners should compensate.

31 卷下:
今有雉、兔同籠，上有三十五頭，下九十四足。問雉、兔各幾何？
Now there are pheasants and rabbits in a cage, with thirty-five heads above and ninety-four feet below. Ask: How many pheasants and how many rabbits are there?
答曰：雉二十三。兔一十二。
Answer: Twenty-three pheasants. Twelve rabbits.
術曰：上置三十五頭，下置九十四足。半其足，得四十七。以少減多，再命之，上三除下三，上五除下五。下有一除上一，下有二除上二，即得。
Method: Place thirty-five heads at the top and ninety-four feet at the bottom. Halve the number of feet, getting forty-seven. Subtract the smaller from the larger and then reassign: for every three at the top, subtract three at the bottom; for every five at the top, subtract five at the bottom. For each one below, subtract one above; for each two below, subtract two above; thus you obtain the result.
又術曰：上置頭，下置足。半其足，以頭除足，以足除頭，即得。
Another method: Place the heads at the top and feet at the bottom. Halve the number of feet; subtract the heads from the feet, then subtract the feet from the heads, thus obtaining the result.

32 卷下:
今有九里渠，三寸魚，頭頭相次。問魚得幾何？
Now there is a ditch nine li long and three cun wide; fish are lined up head to tail. Ask: How many fish can fit?
答曰：五萬四千。
Answer: Fifty-four thousand.
術曰：置九里，以三百步乘之，得二千七百步。又以六尺乘之，得一萬六千二百尺。上十之，得一十六萬二千寸。以魚三寸除之，即得。
Method: Place nine li, multiply by three hundred bu to get two thousand seven hundred bu. Then multiply by six chi to get one hundred sixty-two thousand chi. Multiply this result by ten, getting 1620 thousand cun. Divide it by three cun for each fish; thus you obtain the number of fish.

33 卷下:
今有長安、洛陽相去九百里。車輪一匝一丈八尺。欲自洛陽至長安，問輪匝幾何？
Now there are Chang'an and Luoyang, 900 li apart. A cart wheel turns one zha (turn) for every one zhang eight chi. If one wishes to travel from Luoyang to Chang'an, how many wheel turns are needed?
答曰：九萬匝。
Answer: Ninety thousand turns.
術曰：置九百里，以三百步乘之，得二十七萬步。又以六尺乘之，得一百六十二萬尺。以車輪一丈八尺為法。除之，即得。
Method: Place nine hundred li, multiply by three hundred steps to get two hundred seventy thousand steps. Then multiply by six chi, getting one million sixty-two thousand chi. Use the cart wheel's length of one zhang eight chi as the divisor. Divide, and you will get the result.

34 卷下:
今有出門望見九隄。隄有九木，木有九枝，枝有九巢，巢有九禽，禽有九雛，雛有九毛，毛有九色。問各幾何？
Now there is someone who goes out of the gate and sees nine dikes. Each dike has nine trees, each tree has nine branches, each branch has nine nests, each nest has nine birds, each bird has nine chicks, each chick has nine feathers, and each feather has nine colors. Ask: How many of each are there?
答曰：木八十一。枝七百二十九。巢六千五百六十一。禽五萬九千四十九。雛五十三萬一千四百四十一。毛四百七十八萬二千九百六十九。色四千三百四萬六千七百二十一。
Answer: Eighty-one trees. Seven hundred and twenty-nine branches. Six thousand five hundred and sixty-one nests. Fifty-nine thousand four hundred and ninety birds. Five hundred thirty-one thousand four hundred forty-one chicks. Four million seven hundred eighty-two thousand nine hundred sixty-nine feathers. Forty-three million four hundred six thousand seven hundred twenty-one colors.
術曰：置九隄，以九乘之，得木之數。又以九乘之，得枝之數。又以九乘之，得巢之數。又以九乘之，得禽之數。又以九乘之，得雛之數。又以九乘之，得毛之敷。又以九乘之，得色之數。
Method says: Set nine dikes, multiply by nine, to get the number of trees. Multiply again by nine, to get the number của branches. Multiply again by nine, to obtain the number of nests. Multiply again by nine, and you get the number of birds. Multiply again by nine to obtain the number of chicks. Multiply again by nine, get the number of feathers. Multiply again by nine to get the number of colors.

35 卷下:
今有三女，長女五日一歸，中女四日一歸，少女三日一歸。問三女幾何日相會？
Now there are three daughters: the eldest daughter returns home every five days, the middle daughter every four days, and the youngest daughter every three days. Ask: After how many days will the three daughters meet again?
答曰：六十日。
Answer: Sixty days.
術曰：置長女五日、中女四日、少女三日，於右方。各列一算於左方。維乘之，各得所到數：長女十二到，中女十五到，少女二十到。又各以歸日乘到數，即得。
Method says: Place the eldest daughter's five days, the middle daughter's four days, and the youngest daughter's three days on the right side. Place one counter for each on the left side. Multiply them, and you get their respective numbers of arrivals: the eldest daughter arrives twelve times, the middle daughter fifteen times, and the youngest daughter twenty times. Then multiply each return day by the number of arrivals, and you will get it.

36 卷下:
今有孕婦行年二十九，難九月。未知所生？
Now there is a pregnant woman whose age is twenty-nine, giving birth in the ninth month. It is unknown what will be born?
答曰：生男。
Answer: A boy will be born.
術曰：置四十九，加難月，減行年。所餘，以天除一，地除二，人除三，四時除四，五行除五，六律除六，七星除七，八風除八，九州除九。其不盡者，奇則為男，耦則為女。
Method says: Set forty-nine, add the month of childbirth, subtract the age. The remainder is divided by: Heaven subtracts one, Earth subtracts two, Humanity subtracts three, Four Seasons subtract four, Five Elements subtract five, Six Laws subtract six, Seven Stars subtract seven, Eight Winds subtract eight, Nine Provinces subtract nine. If the remainder is not exhausted, an odd number means a boy will be born, and an even number means a girl will be born.

URN: ctp:sunzi-suan-jing/juan-xia