# Archimedes' cattle problem

The Archimedes' cattle problem is a fiendishly hard problem involving very large numbers that Archimedes presented in a 44-line letter to Eratosthenes, the chief librarian at Alexandria. It ran as follows:

"If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colors, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each color, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

"But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colors in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom."

The answer to the first part of the problem – the smallest solution
for the total number of cattle – turns out to be 50,389,082. But when
the extra two constraints in the second part are factored in, the solution
is vastly larger. The approximate answer of 7.76 × 10^{202544} was
found in 1880 by A. Amthor, having reduced the problem to a form called
a Pell equation.^{1} His calculations
were continued by an ad hoc group called the Hillsboro Mathematical Club,
of Hillsboro, Illinois, between 1889 and 1893. The club's three members
(Edmund Fish, George Richards, and A. H. Bell) calculated the first 31 digits
and the last 12 digits of the smallest total number of cattle to be

77602714064868182695302328332__09__ ... 719455081800

though the two underlined digits should be 13.^{2} In 1931, a correspondent
to the *New York Times* wrote: "Since it has been calculated that
it would take the work of a thousand men for a thousand years to determine
the complete [exact] number [of cattle], it is obvious that the world will
never have a complete solution." But "obvious" and "never" are words designed
to make a fool of prognosticators. Enter the computer! In 1965, with the
help of an IBM 7040, H. C. Williams, R. A. German, and C. R. Zarnke reported
a complete solution to the Cattle Problem, though it was 1981 before all
202545 digits were published,^{3} by Harry Nelson, who used a Cray-1
supercomputer to generate the answer, which begins:

7.760271406486818269530232833213... × 10^{202544}.

### References

1. Amthor, A. and Krumbiegel, B. "Das Problema bovinum des Archimedes." *Z. Math. Phys.*, 25: 121-171 (1880).

2. Bell, A. H. "'Cattle Problem.' By Archimedes 251 B.C.'" *Amer.
Math. Monthly*, 2: 140 (1895).

3. Vardi, I. "Archimedes' Cattle Problem." *Amer. Math. Monthly*,
105: 305-319 (1998).