Chapter 45 Chapter 3: The Hundred Chickens Problem

"Zhang Qiu Jian Suan Jing" also raised an indeterminate problem, that is, the famous hundred-chicken problem in the history of mathematics in the world: "Today, there is a rooster that always costs five, a hen that costs three, and a chick that costs one. Hundreds. Ask the geometry of the rooster, the mother, and the chicks?" According to the art, the equation is listed: "Zhang Qiu Jian Suan Jing" recognizes that this is an indeterminate equation, and gives (4, 18, 78), (8, 11, 81), (12, 4, 84) three sets of solutions, which are all positive integers untie. "Zhang Qiu Jian Suan Jing" hints how to understand: "The technique says: every time a rooster increases by four, every time a hen loses seven, and every time a chick gains three, you get it."Qian Baocong’s understanding is: Multiply the second line by 3, subtract the first line, and turn it into 7x+4y=100, where 4y and 100 are both multiples of 4, so x should be a multiple of 4: x=4t, then y =25-7t, z=75+3t, let t=1, 2, 3, then x=4, 8, 12, y=18, 11, 4, z=78, 81, 84.Because a positive integer solution must be found, x cannot be 0 or a negative number, nor can it be greater than 12. There can only be the above three sets of solutions.Later, people did not find a general solution to the Hundred Chickens Problem. It was not until the middle of the 19th century, after the revival of mathematics in the Song and Yuan Dynasties, that Luo Tengfeng's "Yi You Lu" and Shi Yuechun's "Hundred Chickens Shu Yan" used Dayan to find a general solution. solution.

The Hundred Chickens Problem had a huge impact on Arabic and European mathematics. The "Book of Algorithms" written by Fibonacci in Italy in the 13th century and the "Key of Arithmetic" written by Al Qassi in Arabia in the 15th century all have the chicken problem, which obviously originated from China.