# Hundred Fowls Problem

The **Hundred Fowls Problem** is a problem first discussed in the fifth century CE Chinese mathematics text *Zhang Qiujian suanjing* (The Mathematical Classic of Zhang Qiujian), a book of mathematical problems written by Zhang Qiujian. It is one of the best known examples of indeterminate problems in the early history of mathematics.^{[1]} The problem appears as the final problem in *Zhang Qiujian suanjing* (Problem 38 in Chapter 3). However, the problem and its variants have appeared in the medieval mathematical literature of India, Europe and the Arab world.^{[2]}

The name "Hundred Fowls Problem" is due to the Belgian historian Louis van Hee.^{[3]}

## Problem statement[edit]

The Hundred Fowls Problem as presented in *Zhang Qiujian suanjing* can be translated as follows:^{[4]}

- "Now one cock is worth 5 qian, one hen 3 qian and 3 chicks 1 qian. It is required to buy 100 fowls with 100 qian. In each case, find the number of cocks, hens and chicks bought."

## Mathematical formulation[edit]

Let *x* be the number of cocks, *y* be the number of hens, and *z* be the number of chicks, then the problem is to find *x*, *y* and *z* satisfying the following equations:

*x*+*y*+*z*= 100- 5
*x*+ 3*y*+*z*/3 = 100

Obviously, only non-negative integer values are acceptable. Expressing *y* and *z* in terms of *x* we get

*y*= 25 − (7/4)*x**z*= 75 + (3/4)*x*

Since *x*, *y* and *z* all must be integers, the expression for *y* suggests that *x* must be a multiple of 4. Hence the general solution of the system of equations can be expressed using an integer parameter *t* as follows:^{[5]}

*x*= 4*t**y*= 25 − 7*t**z*= 75 + 3*t*

Since *y* should be a non-negative integer, the only possible values of *t* are 0, 1, 2 and 3. So the complete set of solutions is given by

- (
*x*,*y*,*z*) = (0,25,75), (4,18,78), (8,11,81), (12,4,84).

of which the last three have been given in *Zhang Qiujian suanjing*.^{[3]} However, no general method for solving such problems has been indicated, leading to a suspicion of whether the solutions have been obtained by trial and error.^{[1]}

The Hundred Fowls Problem found in *Zhang Qiujian suanjing* is a special case of the general problem of finding integer solutions of the following system of equations:

*x*+*y*+*z*=*d**ax*+*by*+*cz*=*d*

Any problem of this type is sometime referred to as "Hundred Fowls problem".^{[3]}

## Variations[edit]

Some variants of the Hundred Fowls Problem have appeared in the mathematical literature of several cultures.^{[1]}^{[2]} In the following we present a few sample problems discussed in these cultures.

### Indian mathematics[edit]

Mahavira's *Ganita-sara-sangraha* contains the following problem:

- Pigeons are sold at the rate of 5 for 3, sarasa-birds at the rate of 7 for 5, swans at the rate of 9 for 7, and peacocks at the rate of 3 for 9 (
*pana*s). A certain man was told to bring 100 birds for 100*pana*s. What does he give for each of the various kinds of birds he buys?

The Bakshali manuscript gives the problem of solving the following equations:

*x*+*y*+*z*= 20- 3
*x*+ (3/2)*y*+ (1/2)*z*= 20

### Medieval Europe[edit]

The English mathematician Alcuin of York (8th century, c.735-19 May 804 AD) has stated seven problems similar to the Hundred Fowls Problem in his *Propositiones ad acuendos iuvenes*. Here is a typical problem:

- If 100 bushels of corn be distributed among 100 people such that each man gets 3 bushels, each woman 2 bushels and each child half a bushel, then how many men, women and children were there?

### Arabian mathematics[edit]

Abu Kamil (850 - 930 CE) considered non-negative integer solutions of the following equations:

*x*+*y*+*z*= 100- 3
*x*+ (/20)*y*+ (1/3)*z*= 100.

## References[edit]

- ^
^{a}^{b}^{c}Victor J. Katz, Annette Imhausen (Editors) (2007).*The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook*. Princeton University Press. p. 307. ISBN 9780691114859.CS1 maint: extra text: authors list (link) - ^
^{a}^{b}Kangshen Shen; John N. Crossley; Anthony Wah-Cheung Lun; Hui Liu (1999).*The Nine Chapters on the Mathematical Art: Companion and Commentary*. Oxford University Press. pp. 415–420. ISBN 9780198539360. - ^
^{a}^{b}^{c}Jean-Claude Martzloff (1997).*A History of Chinese Mathematics*. Berlin: Springer-verlag. pp. 307–309. **^**Lam Lay Yong (September 1997). "Zhang Qiujian Suanjing (The Mathematical Classic of Zhang Qiujian). An Overview".*Archive for History of Exact Sciences*.**50**(34): 201–240. JSTOR 41134109.**^**Oystein Ore (2012).*Number Theory and its History*. Courier Corporation. pp. 116–141. ISBN 9780486136431.