《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§20. Results of multiplication & division: multiples of 4

§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 = 4.

Translation

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

Source text Target text Notes
四四一十六、自相乘、得二百五十六。四人分之、人得六十四。 Four fours [are] sixteen, [which], multiplied with itself, resulteth in two hundred [and] fifty-six. [With] four people sharing it, [each] person getteth sixty-four.
三四一十二、自相乘、得一百四十四。三人分之、人得四十八。 Three fours [are] twelve, [which], multiplied with itself, resulteth in one hundred [and] forty-four. [With] three people sharing it, [each] person getteth forty-eight.
二四如八、自相乘、得六十四。二人分之、人得三十二。 Two fours [are] as eight, [which], multiplied with itself, resulteth in sixty-four. [With] two people sharing it, [each] person getteth thirty-two.
一四如四、自相乘、得一十六。一人得一十六。 One four [is] as four, [which], multiplied with itself, resulteth in sixteen. One person getteth sixteen.
右四四一條、得四十、自相乘、得一千六百。四人分之、人得四百。 [The] four fours above [as] one strand, result in forty, [which], multiplied with itself, resulteth in one thousand six hundred. [With] four people sharing it, [each] person getteth four hundred.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 4 \times 4 + 3 \times 4 + 2 \times 4 + 1 \times 4 = 40; \\ 40 \times 40 = 1600; \quad 1600 \div 4 = 400. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §20". <https://yawnoc.github.io/sun-tzu/i/20> Accessed yyyy-mm-dd.