《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§26. A system of linear equations (1)

This section gives the solution to a specific system of linear equations.

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
今有甲乙丙三人持錢。 Suppose there be three people A, B, [and] C holding coins.
甲語乙丙、各將公等所持錢半以益我錢、成九十。 A telleth B [and] C, [If] each take [the] coins that [my] lords hold [and] halve [them] to benefit my coins, [they] become ninety.
  • 語： tell

    語、去聲、 Cantonese: yü6, Mandarin: yù

乙復語甲丙、各將公等所持錢半以益我錢、成七十。 B again telleth A [and] C, [If] each take [the] coins that [my] lords hold [and] halve [them] to benefit my coins, [they] become seventy.
丙復語甲乙、各將公等所持錢半以益我錢、成五十六。 C again telleth A [and] B, [If] each take [the] coins that [my] lords hold [and] halve [them] to benefit my coins, [they] become fifty-six.
問三人元持錢各幾何。 [We] ask, how many coins each held [the] three people originally?
  • Version B has 若干 for 幾何.
  • In modern notation, we have the system of linear equations
    \begin{aligned} a + \frac{b + c}{2} &= T_A \\[\tallspace] b + \frac{c + a}{2} &= T_B \\[\tallspace] c + \frac{a + b}{2} &= T_C, \end{aligned}
    where T_A = 90, T_B = 70, and T_C = 56. The text to follow gives the solution as
    \begin{aligned} a &= \frac{3T_A}{2} - \frac{T_B}{2} - \frac{T_C}{2} = 72 \\[\tallspace] b &= \frac{3T_B}{2} - \frac{T_C}{2} - \frac{T_A}{2} = 32 \\[\tallspace] c &= \frac{3T_C}{2} - \frac{T_A}{2} - \frac{T_B}{2} = 4, \end{aligned}
    but there is no derivation of this (which is rather unsatisfactory).
答曰、甲七十二、乙三十二、丙四。 Answer saith: A seventy-two; B thirty-two; C four.
術曰、先置三人所語為位。 Method saith: first put [down] that told [by the] three people as places.
以三乘之、各為積、甲得二百七十、乙得二百一十、丙得一百六十八。 Multiplying them by three, each becometh [a] product: A resulteth in two hundred [and] seventy; B resulteth in two hundred [and] ten; C resulteth in one hundred [and] sixty-eight.
各半之、甲得一百三十五、乙得一百五、丙得八十四。 Halving them each: A resulteth in one hundred [and] thirty-five; B resulteth in one hundred [and] five; C resulteth in eighty-four.
又置甲九十、乙七十、丙五十六、各半之。 And put [down] A's ninety, B's seventy, [and] C's fifty-six, [and] halve them each.
以甲乙減丙、以甲丙減乙、以乙丙減甲、即各得元數。 Subtracting of C by A [and] B, subtracting of B by A [and] C, [and] subtracting of A by B [and] C, doth each result in [its] original number.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §26". <https://yawnoc.github.io/sun-tzu/ii/26> Accessed yyyy-mm-dd.