《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§17. Results of multiplication & division: multiples of 7

§15 through §22 give the results of multiplications and divisions of the form

\begin{aligned} (m n) \times (m n) &= m^2 n^2 \\ (m^2 n^2) \div m &= m n^2, \end{aligned}

along with the result of the division

\Bigl[ n \times n + (n - 1) \times n + \dots + 1 \times n \Bigr]^2 \div n.

Since this gets rather repetitive and boring, you may wish to skip to §23.

This section corresponds to n = 7.

Translation

Chinese source text: Version B, Version C, Version D. (Missing in Version A up to 四人分之, where it resumes.)
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
七七四十九、自相乘、得二千四百一。七人分之、人得三百四十三。 Seven sevens [are] forty-nine, [which], multiplied with itself, resulteth in two thousand four hundred [and] one. [With] seven people sharing it, [each] person getteth three hundred [and] forty-three.
六七四十二、自相乘、得一千七百六十四。六人分之、人得二百九十四。 Six sevens [are] forty-two, [which], multiplied with itself, resulteth in one thousand seven hundred [and] sixty-four. [With] six people sharing it, [each] person getteth two hundred [and] ninety-four.
五七三十五、自相乘、得一千二百二十五。五人分之、人得二百四十五。 Five sevens [are] thirty-five, [which], multiplied with itself, resulteth in one thousand two hundred [and] twenty-five. [With] five people sharing it, [each] person getteth two hundred [and] forty-five.
四七二十八、自相乘、得七百八十四。四人分之、人得一百九十六。 Four sevens [are] twenty-eight, [which], multiplied with itself, resulteth in seven hundred [and] eighty-four. [With] four people sharing it, [each] person getteth one hundred [and] ninety-six.
三七二十一、自相乘、得四百四十一。三人分之、人得一百四十七。 Three sevens [are] twenty-one, [which], multiplied with itself, resulteth in four hundred [and] forty-one. [With] three people sharing it, [each] person getteth one hundred [and] forty-seven.
二七十四、自相乘、得一百九十六。二人分之、人得九十八。 Two sevens [are] fourteen, [which], multiplied with itself, resulteth in one hundred [and] ninety-six. [With] two people sharing it, [each] person getteth ninety-eight.
  • Version B has 二七一十四 for 二七十四.
一七如七、自相乘、得四十九。一人得四十九。 One seven [is] as seven, [which], multiplied with itself, resulteth in forty-nine. One person getteth forty-nine.
右七七一條、得一百九十六、自相乘、得三萬八千四百一十六。七人分之、人得五千四百八十八。 [The] seven sevens above [as] one strand, result in one hundred [and] ninety-six, [which], multiplied with itself, resulteth in three myriad eight thousand four hundred [and] sixteen. [With] seven people sharing it, [each] person getteth five thousand four hundred [and] eighty-eight.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 7 \times 7 + 6 \times 7 + \dots + 1 \times 7 = 196; \\ 196 \times 196 = 38416; \quad 38416 \div 7 = 5488. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §17". <https://yawnoc.github.io/sun-tzu/i/17> Accessed yyyy-mm-dd.