《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§21. Results of multiplication & division: multiples of 3

§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 = 3.

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
三三如九、自相乘、得八十一。三人分之、人得二十七。 Three threes [are] as nine, [which], multiplied with itself, resulteth in eighty-one. [With] three people sharing it, [each] person getteth twenty-seven.
二三如六、自相乘、得三十六。二人分之、人得一十八。 Two threes [are] as six, [which], multiplied with itself, resulteth in thirty-six. [With] two people sharing it, [each] person getteth eighteen.
一三如三、自相乘、得九。一人得九。 One three [is] as three, [which], multiplied with itself, resulteth in nine. One person getteth nine.
右三三一條、得一十八、自相乘、得三百二十四。三人分之、人得一百八。 [The] three threes above [as] one strand, result in eighteen, [which], multiplied with itself, resulteth in three hundred [and] twenty-four. [With] three people sharing it, [each] person getteth one hundred [and] eight.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 3 \times 3 + 2 \times 3 + 1 \times 3 = 18; \\ 18 \times 18 = 324; \quad 324 \div 3 = 108. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §21". <https://yawnoc.github.io/sun-tzu/i/21> Accessed yyyy-mm-dd.