今有人盜庫絹、不知所失幾何。但聞草中分絹、人得六匹、盈六匹、人得七匹、不足七匹。 |
Suppose there be people [having] stolen warehoused strong-silk,
[and we] know not how much that lost [be].
But [we] hear [a] sharing of [the] strong-silk amidst [the] grass:
[each] person getting six lengths,
[be there] excess six lengths;
[each] person getting seven lengths,
[be there] not sufficient seven lengths. |
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問人絹各幾何。 |
[We] ask, how much each [be the] people [and the] strong-silk? |
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Version A and Version D have 得幾何.
I have taken 各幾何 per Version B and Version C.
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In modern notation,
suppose the length of strong-silk is l, stolen by p people,
with surplus Ys=6lengths
if each person takes Xs=6lengths,
and deficit Yd=7lengths
if each person takes Xd=7lengths.
Then
l−pXsl−pXd=+Ys=−Yd.
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答曰、賊一十三人、絹八十四匹。 |
Answer saith: [the] thieves [be] thirteen people,
[and the] strong-silk eighty-four lengths. |
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術曰、先置人得六匹於右上、盈六匹於右下、後置人得七匹於左上、不足七匹於左下。 |
Method saith:
first put [each] person getting six lengths upon [the] upper right,
[and the] excess six lengths upon [the] lower right;
[and] afterward put [each] person getting seven lengths upon [the] upper left,
[and the] not sufficient seven lengths upon [the] lower left. |
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維乘之、所得、并之、為絹。 |
[In] linkage multiply them;
[of] those which result, combining them, maketh [the] strong-silk. |
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Having formed the matrix
(XdYdXsYs),
we are told to form the products
XsYd and XdYs,
and add them together to obtain l.
This does not make dimensional sense; area cannot equal length.
The modern equivalent is to apply Cramer's rule to the system above,
which yields
l=∣∣∣∣∣11−Xs−Xd∣∣∣∣∣∣∣∣∣∣+Ys−Yd−Xs−Xd∣∣∣∣∣=Xd−XsXsYd+XdYs=7−66×7+7×6lengthslengths2.
We see that Sun Tzŭ's prescription only works
because the denominator is Xd−Xs=7lengths−6lengths=1length,
so the division by Xd−Xs can be omitted
if all computations are done in units of length (匹).
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并下盈不足、為人。 |
Combining [the] excess [and the] not sufficient below, maketh [the] people. |
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We are told that the sum Ys+Yd
forms the number of people.
Again this does not make dimensional sense,
and it only works because Xd−Xs=7lengths−6lengths=1length.
Indeed Cramer's rule gives
p=∣∣∣∣∣11−Xs−Xd∣∣∣∣∣∣∣∣∣∣11+Ys−Yd∣∣∣∣∣=Xd−XsYs+Yd=7−66+7lengthslengths
for the number of people.
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