Archimedes' Cattle Problem

Also called the Bovinum Problema. It is stated as follows: ``The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?''

Solution consists of solving the simultaneous Diophantine Equations in Integers $W$, $X$, $Y$, $Z$ (the number of white, black, spotted, and brown bulls) and $w$, $x$, $y$, $z$ (the number of white, black, spotted, and brown cows),

$$W = {\textstyle{5\over 6}} X+Z$$ (1)

$$X = {\textstyle{9\over 20}} Y+Z$$ (2)

$$Y = {\textstyle{13\over 42}}W+Z$$ (3)

$$w = {\textstyle{7\over 12}}(X+x)$$ (4)

$$x = {\textstyle{9\over 20}}(Y+y)$$ (5)

$$y = {\textstyle{11\over 30}}(Z+z)$$ (6)

$$z = {\textstyle{13\over 42}}(W+w).$$ (7)

The smallest solution in Integers is

$$W = 10{,}366{,}482$$ (8)

$$X = \phantom{z}7{,}460{,}514$$ (9)

$$Y = \phantom{z}7{,}358{,}060$$ (10)

$$Z = \phantom{z}4{,}149{,}387$$ (11)

$$w = \phantom{z}7{,}206{,}360$$ (12)

$$x = \phantom{z}4{,}893{,}246$$ (13)

$$y = \phantom{z}3{,}515{,}820$$ (14)

$$z = \phantom{z}5{,}439{,}213.$$ (15)

A more complicated version of the problem requires that $W+X$ be a Square Number and $Y+Z$ a Triangular Number. The solution to this Problem are numbers with 206544 or 206545 digits.

References

Amthor, A. and Krumbiegel B. ``Das Problema bovinum des Archimedes.'' Z. Math. Phys. 25, 121-171, 1880.

Archibald, R. C. ``Cattle Problem of Archimedes.'' Amer. Math. Monthly 25, 411-414, 1918.

Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, pp. 249-252, 1966.

Bell, A. H. ``Solution to the Celebrated Indeterminate Equation $x^2 - ny^2 = 1$.'' Amer. Math. Monthly 1, 240, 1894.

Bell, A. H. ```Cattle Problem.' By Archimedes 251 BC.'' Amer. Math. Monthly 2, 140, 1895.

Bell, A. H. ``Cattle Problem of Archimedes.'' Math. Mag. 1, 163, 1882-1884.

Dörrie, H. ``Archimedes' Problema Bovinum.'' §1 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 3-7, 1965.

Grosjean, C. C. and de Meyer, H. E. ``A New Contribution to the Mathematical Study of the Cattle-Problem of Archimedes.'' In Constantin Carathéodory: An International Tribute, Vols. 1 and 2 (Ed. T. M. Rassias). Teaneck, NJ: World Scientific, pp. 404-453, 1991.

Merriman, M. ``Cattle Problem of Archimedes.'' Pop. Sci. Monthly 67, 660, 1905.

Rorres, C. ``The Cattle Problem.'' http://www.mcs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html.

Vardi, I. ``Archimedes' Cattle Problem.'' Amer. Math. Monthly 105, 305-319, 1998.