《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§31. A system of linear equations (4)

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
今有雉兔同籠、上有三十五頭、下有九十四足。問雉兔各幾何。 Suppose there be pheasants [and] rabbits [in the] same cage: above be there thirty-five heads; below be there ninety-four feet. [We] ask, how many each [be the] pheasants [and the] rabbits?
  • Version C is erroneously missing 有 in 下有.
  • In modern notation, we have the system
    \begin{aligned} p + r &= 35 && \text{(heads)} \\ 2p + 4r &= 94 && \text{(feet)} \end{aligned}
    in p the number of pheasants and r the number of rabbits.
答曰、雉二十三、兔一十二。 Answer saith: [the] pheasants twenty-three, [and the] rabbits twelve.
術曰、上置三十五頭、下置九十四足。半其足、得四十七。 Method saith: put above [the] thirty-five heads, [and] put below [the] ninety-four feet. Halving [the] feet, resulteth in forty-seven.
  • In modern notation, we begin with the augmented matrix
    \roundbr{ \begin{array}{cc|c} p & r & 35 \\ 2p & 4r & 94 \end{array} },
    where we have written p and r explicitly for readability. Upon halving the bottom row (for feet) we obtain
    \roundbr{ \begin{array}{cc|c} p & r & 35 \\ p & 2r & 47 \end{array} }.
以少減多、再命之、 Subtracting of [the] greater by [the] lesser, [and] again commanding it:
上三除下三、上五除下五、下有一除上一、下有二除上二、即得。 [for the] three above removing three below, [and for the] five above removing five below; [for] there being one below removing one above, [and for] there being two below removing two above: [we] are done.
  • The first subtraction 47 - 35 = 12 determines the number of rabbits:
    \roundbr{ \begin{array}{cc|c} p & r & 35 \\ & r & 12 \end{array} }.
  • The second subtraction 35 - 12 = 23 determines the number of pheasants:
    \roundbr{ \begin{array}{cc|c} p & & 23 \\ & r & 12 \end{array} }.
又術曰、上置頭、下置足。半其足、以頭除足、以足除頭、即得。 Also method saith: put above [the] heads, [and] put below [the] feet. Halve [the] feet; removing from [the] feet by [the] heads, [and] removing from [the] heads by [the] feet, [we] are done.
  • This is a restatement of the above method, but without going into the details of the individual digits being subtracted.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §31". <https://yawnoc.github.io/sun-tzu/iii/31> Accessed yyyy-mm-dd.