# China

### Mathematician

Sun Zi was a scholar who wrote the Sunzi Suanjing text which covered measurement, the abacus, two systems for designating high powers of ten, fractions, areas, and volumes. In his text, Zi posed the earliest known occurrence of the Chinese remainder theorem.

### Problem 1

Suppose that, after going through a town gate, you see 9 dykes, with 9 trees on each dyke, 9 branches on each tree, 9 nests on each branch, and 9 birds in each nest, where each bird has 9 fledglings and each fledgling has 9 feathers with 9 different colors in each feather. How many are there of each?

### Problem 2

Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 3 are left over, and when counted in sevens, 2 are left over. How many objects are there?

### Mathematician

In 1261 Yang Hui wrote a Detailed analysis of the Nine Chapters on the Mathematical Art , the classical Chinese handbook of mathematics. The handbook contains 246 problems with practical applications to engineering, surveying, trade, and taxation. Yang analyzed each problem logically, numerically, and practically. In addition, he explored other areas of mathematics: Pascal's triangle (up to the sixth row) and the natural numbers (sum of the squares from m 2 to ). Hui also proved that and contributed to magic squares (, , , , , , , and ).

### Problem 1

Now 1 cubic cun of jade weighs 7 liang, and 1 cubic cun of rock weighs 6 liang. Now there is a cube of side 3 cun consisting of a mixture of jade and rock which weighs 11 jin. What are the weights of jade and rock in the cube [note 1 jin = 16 liang]?

### Problem 2

100 coins buy Wenzhou oranges, green oranges, and golden oranges, 100 in total. If a Wenzhou orange costs 7 coins, a green orange 3 coins, and 3 golden oranges cost 1 coin, how many oranges of the three kinds will be bought?

### Mathematician

Little is known about Zhang Qiujian, a learned teacher, except that he wrote a Mathematical Manual in which he outlined the calculation of square and cube roots, solving of equations and systems of equations, sum of arithmetic series, proportions and fractions, and geometry. Among Qiujian's solutions to the problems analyzed in his textbook, he solves systems of linear equations using a method developed by Gauss (Gaussian elimination).

### Problem 1

A circular road around a hill is 325 li long. Three persons A , B , and C run along the road. A runs 150 li per day, B runs 120 li per day, and C runs 90 li per day. If they start at the same time from the same place, after how many days will they meet again.

### Problem 2

There are three persons, A , B , and C each with a number of coins. A says “If I take ? of B 's coins and ? of C 's coins then I hold 100?. B says “If I take ? of A 's coins and ½ of C 's coins then I hold 100 coins”. C says “If I take ? of A 's coins and ? of B 's coins, then I hold 100 coins”. Tell me how many coins do A , B , and C hold?

### Problem 3

Cockerels costs 5 qian each, hens 3 qian each and three chickens cost 1 qian. If 100 fowls are bought for 100 qian, how many cockerels, hens and chickens are there?

### Mathematician

Zhu Shijie wrote two textbooks: True Reflections of the Four Unknowns and Introduction to Mathematical Studies. In the latter, Shijie discusses the rule of three, areas and volumes, and the rule of false double position. Shijie also developed a method to solve simultaneous equations very similar to Gauss'. In True Reflections of the Four Unknowns, Shijie includes “the table of the ancient method of powers up to the eighth” (Pascal's triangle), a method for solving equations up to degree fourteen which was later rediscovered by two mathematicians in England and Italy (Horner and Ruffini), formulas for the sum of an infinite series, and 288 problems. In giving his solutions to these problems, Shijie did not give the most efficient method, but rather introduced complicated methods to highlight interesting aspects of the problems.

### Problem 1

A right-angled triangle has area 30 bu. The sum of the base and height of the triangle is 17 bu. What is the sum of the base and the hypotenuse?

### Problem 2

Given the relations and between the sides of a right angled triangle x , y , z where z is the hypotenuse. Determine .

### Problem 3

Let d be the diameter of the circle inscribed in a right triangle (you should use the relation where 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 and . Determine y .

### Mathematician

In his youth Cheng Da Wei was academically gifted, but although he was well versed in scholarly matters, he continued to carry out his profession as a sincere Local Agent, without becoming a scholar. He never lagged behind either on the classics or on ancient writings with old style characters, but was particularly gifted in arithmetic. In the prime of his life he visited the fairs of Wu and Chu. When he came across books that talked about “square fields” or “grain with the husk removed” … he never looked at the price before purchasing them. He questioned respectable old men who were experienced in the practice of arithmetic and gradually and indefatigably formed his own collection of difficult problems.

### Problem 1

Boy shepherd B with his one sheep behind him asked shepherd A “Are there 100 sheep in your flock?”. Shepherd A replies “Yet add the same flock, the same flock again, half, one quarter flock and your sheep. There are then 100 sheep altogether.” How many sheep is in shepherd A's flock?

J-C Martzloff, A history of Chinese mathematics (Berlin-Heidelberg, 1997).
J-C Martzloff, Histoire des mathématiques chinoises (Paris, 1987).
K Takeda, The characteristics of Chinese mathematics in the Ming dynasty (Japanese), J. Hist. Sci. Tokyo 29 (1954), 8-18.