《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§15. Results of multiplication & division: multiples of 9

§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 = 9.

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
八九七十二、自相乘、得五千一百八十四。八人分之、人得六百四十八。 Eight nines [are] seventy-two, [which], multiplied with itself, resulteth in five thousand one hundred [and] eighty-four. [With] eight people sharing it, [each] person getteth six hundred [and] forty-eight.
  • Version A erroneously has 得 for 人得.
  • Version C erroneously has 六百四十八五 for 六百四十八.
七九六十三、自相乘、得三千九百六十九。七人分之、人得五百六十七。 Seven nines [are] sixty-three, [which], multiplied with itself, resulteth in three thousand nine hundred [and] sixty-nine. [With] seven people sharing it, [each] person getteth five hundred [and] sixty-seven.
  • Version A erroneously has 得 for 人得.
  • Version C erroneously has 五百六十七五 for 五百六十七.
六九五十四、自相乘、得二千九百一十六。六人分之、人得四百八十六。 Six nines [are] fifty-four, [which], multiplied with itself, resulteth in two thousand nine hundred [and] sixteen. [With] six people sharing it, [each] person getteth four hundred [and] eighty-six.
五九四十五、自相乘、得二千二十五。五人分之、人得四百五。 Five nines [are] forty-five, [which], multiplied with itself, resulteth in two thousand [and] twenty-five. [With] five people sharing it, [each] person getteth four hundred [and] five.
四九三十六、自相乘、得一千二百九十六。四人分之、人得三百二十四。 Four nines [are] thirty-six, [which], multiplied with itself, resulteth in one thousand two hundred [and] ninety-six. [With] four people sharing it, [each] person getteth three hundred [and] twenty-four.
  • Version A is missing four whole pages after 一千二百九十六, from 四人分之 in the line above until 四人分之 after 四七二十八 in §17. These missing pages correspond to pages 21 through 24 in Version D, which up to this point has the same page breaks as Version A.
三九二十七、自相乘、得七百二十九。三人分之、人得二百四十三。 Three nines [are] twenty-seven, [which], multiplied with itself, resulteth in seven hundred [and] twenty-nine. [With] three people sharing it, [each] person getteth two hundred [and] forty-three.
二九一十八、自相乘、得三百二十四。二人分之、人得一百六十二。 Two nines [are] eighteen, [which], multiplied with itself, resulteth in three hundred [and] twenty-four. [With] two people sharing it, [each] person getteth one hundred [and] sixty-two.
一九如九、自相乘、得八十一。一人得八十一。 One nine [is] as nine, [which], multiplied with itself, resulteth in eighty-one. One person getteth eighty-one.
右九九一條、得四百五、自相乘、得一十六萬四千二十五。九人分之、人得一萬八千二百二十五。 [The] nine nines above [as] one strand, result in four hundred [and] five, [which], multiplied with itself, resulteth in sixteen myriad four thousand [and] twenty-five. [With] nine people sharing it, [each] person getteth one myriad eight thousand two hundred [and] twenty-five.
  • 右： above; lit. aright

    In English, preceding material lies above, since English is written left-to-right, top-to-bottom.
    In Chinese, preceding material lies to the right, since (at least classically) Chinese is written top-to-bottom, right-to-left.

  • Version C erroneously has 二萬八千二百二十五 for 一萬八千二百二十五.
  • 條, "strand", is the classifier for problem/question. I think 右九九一條, "[the] nine nines above [as] one strand", is saying "considering 9 \times 9 through 1 \times 9 above, together in one single problem", i.e.
    \begin{gathered} 9 \times 9 + 8 \times 9 + \dots + 1 \times 9 = 405; \\ 405 \times 405 = 164025; \quad 164025 \div 9 = 18225. \end{gathered}
    Note that "nine nines above" includes 9 \times 9 from §13. (The source text has no sectioning, only strategically placed line breaks.)

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §15". <https://yawnoc.github.io/sun-tzu/i/15> Accessed yyyy-mm-dd.