156 6. CALCULATIONan algorithm amounts to the drama that results when these roles are acted. That
is why it is so important that each part of the algorithm have its own name. The
letters that we use for variables amount to names assigned to roles in the drama.
A declaration of variables at the beginning of a program is analogous to the section
that used to be titled "Dramatis Persona?" at the beginning of a play.
Explain long division from this point of view, using the roles of dividend, divi-
sor, quotient, and remainder.
6.17. Imitate the reasoning used in solving the problem of riders and carts above
to solve Problem 17 of the Sun Zi Suan Jing. The problem asks how many guests
were at a banquet if every two persons shared a bowl of rice, every three persons
a bowl of soup, and every four persons a bowl of meat, leading to a total of 65
bowls. Don't use algebra, but try to explain the rather cryptic solution given by
Sun Zi: Put down 65 bowls, multiply by 12 to obtain 780, and divide by 13 to get
the answer.
6.18. Compare the following loosely interpreted problems from the Jiu Zhang Suan-
shu and the Ahmose Papyrus. First, from the Jiu Zhang Suanshu: Five officials
went hunting and killed five deer. Their ranks entitle them to shares in the pro-
portion 1:2:3:4:5. What part of a deer does each receive?
Second, from the Ahmose Papyrus (Problem 40): 100 loaves of bread are to
be divided among five people (in arithmetic progression), in such a way that the
amount received by the last two (together) is one-seventh of the amount received
by the first three (together). How much bread does each person receive?
6.19. Compare the interest problem (Problem 20 of Chapter 3) from the Jiu Zhang
Suanshu discussed above, with the following problem, taken from the American
textbook New Practical Arithmetic by Benjamin Greenleaf (1876):
The interest on $200 for 4 months being $4, what will be the in-
terest on $590 for 1 year and 3 months?
Are there any significant differences at all in the nature of the two problems, written
nearly 2000 years apart?
6.20. Problem 4 in Chapter 6 of the Jiu Zhang Suanshu involves what is called
double false position. The problem reads as follows: A number of families contribute
equal amounts to purchase a herd of cattle. If the contribution (the same for each
family) were such that seven families contribute a total of 190 [units of money],
there would be a deficit of 330 [units of money]; but if the contribution were such
that nine families contribute 270 [units of money], there would be a surplus of 30
[units of money]. Assuming that the families each contribute the correct amount,
how much does the herd cost, and how many families are involved in the purchase?
Explain the solution given by the author of the Jiu Zhang Suanshu, which goes
as follows. Put down the proposed values (assessment to each family, that is,
and ™ = 30), and below each put down the corresponding surplus or deficit (a
positive number in each case). Cross-multiply and add the products to form the
shi (30 • ø + 330 · 2709 = º&Ì). Add the surplus and deficit to form the fa
(330 + 30 = 360). Subtract the smaller of the proposed values from the larger, to
get the difference (™ - = ø). Divide the shi by the difference to get the cost
of the goods (^fjp = 3750); divide the fa by the difference to get the number of
families = 126).