《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§16. Results of multiplication & division: multiples of 8

§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 = 8.

Translation

Chinese source text: Version B, Version C, Version D. (Missing in Version A.)
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
八八六十四、自相乘、得四千九十六。八人分之、人得五百十二。 Eight eights [are] sixty-four, [which], multiplied with itself, resulteth in four thousand [and] ninety-six. [With] eight people sharing it, [each] person getteth five hundred [and] twelve.
  • Version B has 五百一十二 for 五百十二.
七八五十六、自相乘、得三千一百三十六。七人分之、人得四百四十八。 Seven eights [are] fifty-six, [which], multiplied with itself, resulteth in three thousand one hundred [and] thirty-six. [With] seven people sharing it, [each] person getteth four hundred [and] forty-eight.
六八四十八、自相乘、得二千三百四。六人分之、人得三百八十四。 Six eights [are] forty-eight, [which], multiplied with itself, resulteth in two thousand three hundred [and] four. [With] six people sharing it, [each] person getteth three hundred [and] eighty-four.
五八四十、自相乘、得一千六百。五人分之、人得三百二十。 Five eights [are] forty, [which], multiplied with itself, resulteth in one thousand six hundred. [With] five people sharing it, [each] person getteth three hundred [and] twenty.
四八三十二、自相乘、得一千二十四。四人分之、人得二百五十六。 Four eights [are] thirty-two, [which], multiplied with itself, resulteth in one thousand [and] twenty-four. [With] four people sharing it, [each] person getteth two hundred [and] fifty-six.
三八二十四、自相乘、得五百七十六。三人分之、人得一百九十二。 Three eights [are] twenty-four, [which], multiplied with itself, resulteth in five hundred [and] seventy-six. [With] three people sharing it, [each] person getteth one hundred [and] ninety-two.
二八十六、自相乘、得二百五十六。二人分之、人得一百二十八。 Two eights [are] sixteen, [which], multiplied with itself, resulteth in two hundred [and] fifty-six. [With] two people sharing it, [each] person getteth one hundred [and] twenty-eight.
一八如八、自相乘、得六十四。一人得六十四。 One eight [is] as eight, [which], multiplied with itself, resulteth in sixty-four. One person getteth sixty-four.
右八八一條、得二百八十八、自相乘、得八萬二千九百四十四。八人分之、人得一萬三百六十八。 [The] eight eights above [as] one strand, result in two hundred [and] eighty-eight, [which], multiplied with itself, resulteth in eight myriad two thousand nine hundred [and] forty-four. [With] eight people sharing it, [each] person getteth one myriad three hundred [and] sixty-eight.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 8 \times 8 + 7 \times 8 + \dots + 1 \times 8 = 288; \\ 288 \times 288 = 82944; \quad 82944 \div 8 = 10368. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §16". <https://yawnoc.github.io/sun-tzu/i/16> Accessed yyyy-mm-dd.