We give here a collection of Chinese problems which are extracted from various articles in our archive on Chinese mathematics or Chinese mathematicians. Many of the problems have answers given in the corresponding article, and some have a description of the method. Each problem has a reference to the article in which it occurs.

**Problem 1:** See Nine Chapters

A good runner can go100paces while a poor runner covers60paces. The poor runner has covered a distance of100paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.

Boy shepherd B with his one sheep behind him asked shepherd A "Are there100sheep in your flock?". Shepherd A replies "Yet add the same flock, the same flock again, half, one quarter flock and your sheep. There are then100sheep altogether." How many sheep is in shepherd A's flock?

Now1cubic cun of jade weighs7liang, and1cubic cun of rock weighs6liang. Now there is a cube of side3cun consisting of a mixture of jade and rock which weighs11jin. Tell: what are the weights of jade and rock in the cube.[Note1jin =16liang]

Suppose that, after going through a town gate, you see9dykes, with9trees on each dyke,9branches on each tree,9nests on each branch, and9birds in each nest, where each bird has9fledglings and each fledgling has9feathers with9different colours in each feather. How many are there of each?

Problem 6: See Nine ChaptersCertain items are purchased jointly. If each person pays8coins, the surplus is3coins, and if each person gives7coins, the deficiency is4coins. Find the number of people and the total cost of the items.

Problem 7: See Nine ChaptersThere are two piles, one containing9gold coins and the other11silver coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs13units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Problem 8: See Sun ZiThere is a square town of unknown dimensions. There is a gate in the middle of each side. Twenty paces outside the North Gate is a tree. If one leaves the town by the South Gate, walks14paces due south, then walks due west for1775paces, the tree will just come into view. What are the dimensions of the town.

Problem 9: See Nine ChaptersSuppose we have an unknown number of objects. When counted in threes,2are left over, when counted in fives,3are left over, and when counted in sevens,2are left over. How many objects are there?

Problem 10: See Li ZhiA cistern is filled through five canals. Open the first canal and the cistern fills in1^{}/3_{}day; with the second, it fills in1day; with the third, in21^{}/2_{}days; with the fourth, in3days, and with the fifth in5days. If all the canals are opened, how long will it take to fill the cistern?

Problem 11: See Li ZhiGiven a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. Two persons A and B start from the west gate. B walks a distance of256pu eastwards. Then A walks a distance of480pu south before he can see B. Find the diameter of the town.

Problem 12: See Li ZhiGiven a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. Person A leaves the west gate and walks south for480pu. B leaves the east gate and walks straight ahead a distance of16pu, when he just sees A. Find the diameter of the town.

Problem 13: See Qin JiushaoGiven a circular walled city of unknown diameter with four gates, one at each of the four cardinal points.135pu directly out of the south gate is a tree. If one walks15pu out of the north gate and then turns east for a distance of208pu, the tree comes into sight. Find the diameter of the town.

Problem 14: See Li ZhiGiven a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. A tree lies three li north of the northern gate. If one turns and walks eastwards for nine li immediately on leaving the southern gate, the tree just comes into view. Find the circumference and the diameter of the city wall.

Problem 15: See Cheng DaweiA square farm has a circular pond in the centre. The land area is13mou and71^{}/2_{}tenths of a mou. The pond is20pu from the edge. Find the length of the side of the farm and the diameter of the pond.

Problem 16: See Cheng DaweiNow a pile of rice is against the wall with a base circumference60chi and an altitude of12chi. What is the volume? Another pile is at an inner corner, with a base circumference of30chi and an altitude of12chi. What is the volume? Another pile is at an outer corner, with base circumference of90chi and an altitude of12chi. What is the volume?

Problem 17: See Cheng DaweiA small river cuts right across a circular field whose area is unknown. Given the diameter of the field and the breadth of the river find the area of the non-flooded part of the field.

Problem 18: See Zhu ShijieIn the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b =81ken and a + c =72ken. Find a, b, and c.

Problem 19: See Wang XiaotongA right-angled triangle has area30bu. The sum of the base and height of the triangle is17bu. What is the sum of the base and hypotenuse?

Problem 20: See Zhang QiujianLet a right angled triangle have sides a, b, c where c is the hypotenuse. If a times b is seven hundred and six and one fiftieth, and if c is thirty six and nine tenths more than a. What are the values of the three sides.

Problem 21: See Zhang QiujianA circular road around a hill is325li long. Three persons A, B, and C run along the road. A runs150li per day, B runs120li per day, and C runs90li per day. If they start at the same time from the same place, after how many days will they meet again.

Problem 22: See Zhang QiujianThere are three persons, A, B, and C each with a number of coins. A says "If I take2^{}/3_{}of B's coins and1^{}/3_{}of C's coins then I hold100". B says If I take2^{}/3_{}of A's coins and1^{}/2_{}of C's coins then I hold100coins". C says "If I take2^{}/3_{}of A's coins and2^{}/3_{}of B's coins, then I hold100coins". Tell me how many coins do A, B, and C hold?

Problem 23: See Yang HuiCockerels costs5qian each, hens3qian each and three chickens cost1qian. If100fowls are bought for100qian, how many cockerels, hens and chickens are there?

100Problem 24: See Yang Huicoins buy Wenzhou oranges, green oranges, and golden oranges,100in total. If a Wenzhou orange costs7coins, a green orange3coins, and3golden oranges cost1coin, how many oranges of the three kinds will be bought?

Problem 25: See Zhu ShijieA number of pheasants and rabbits are placed together in the same cage. Thirty-five heads and ninety-four feet are counted. Find the number of pheasants and rabbits.

Problem 26: See Zhu ShijieGiven the relations2yz=z^{2}+xzand2x+ 4y+ 4z=x(y^{2}-z+x)between the sides of a right angled triangle x, y, z where z is the hypotenuse, findd= 2x+ 2y.

Problem 27: See Zhu ShijieIf the cube law is applied to the rate of recruiting soldiers and it is found that on the first day3cubed are recruited,4cubed on the second day, and on each succeeding day the cube of a number one greater than the previous day are recruited, how many soldiers in total will have been recruited after15days? How many after n days?

Let d be the diameter of the circle inscribed in a right triangle(you should use the relationd=x+y-zwhere x, y, z are as defined below). Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle. Given that dxy =24and x + z =9find y.

**Article by:** *J J O'Connor* and *E F Robertson*

**MacTutor History of Mathematics**

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Chinese_problems.html]