《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§18. Results of multiplication & division: multiples of 6

§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 = 6.

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
六六三十六、自相乘、得一千二百九十六。六人分之、人得二百一十六。 Six sixes [are] thirty-six, [which], multiplied with itself, resulteth in one thousand two hundred [and] ninety-six. [With] six people sharing it, [each] person getteth two hundred [and] sixteen.
五六三十、自相乘、得九百。五人分之、人得一百八十。 Five sixes [are] thirty, [which], multiplied with itself, resulteth in nine hundred. [With] five people sharing it, [each] person getteth one hundred [and] eighty.
四六二十四、自相乘、得五百七十六。四人分之、人得一百四十四。 Four sixes [are] twenty-four, [which], multiplied with itself, resulteth in five hundred [and] seventy-six. [With] four people sharing it, [each] person getteth one hundred [and] forty-four.
三六一十八、自相乘、得三百二十四。三人分之、人得一百八。 Three sixes [are] eighteen, [which], multiplied with itself, resulteth in three hundred [and] twenty-four. [With] three people sharing it, [each] person getteth one hundred [and] eight.
二六一十二、自相乘、得一百四十四。二人分之、人得七十二。 Two sixes [are] twelve, [which], multiplied with itself, resulteth in one hundred [and] forty-four. [With] two people sharing it, [each] person getteth seventy-two.
一六如六、自相乘、得三十六。一人得三十六。 One six [is] as six, [which], multiplied with itself, resulteth in thirty-six. One person getteth thirty-six.
右六六一條、得一百二十六、自相乘、得一萬五千八百七十六。六人分之、人得二千六百四十六。 [The] six sixes above [as] one strand, result in one hundred [and] twenty-six, [which], multiplied with itself, resulteth in one myriad five thousand eight hundred [and] seventy-six. [With] six people sharing it, [each] person getteth two thousand six hundred [and] forty-six.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 6 \times 6 + 5 \times 6 + \dots + 1 \times 6 = 126; \\ 126 \times 126 = 15876; \quad 15876 \div 6 = 2646. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §18". <https://yawnoc.github.io/sun-tzu/i/18> Accessed yyyy-mm-dd.