In response to Apollonious of Perga’s innocent improvement on Archimedes’ large number system, Archimedes concocted a computational problem about cattle whose solution is so large it was not solved until recently, by the fastest computer in the world (at the time).  The twenty-two hundred year old problem is to determine the number of bulls and cows in each herd, with the following constraints:

1.      White bulls = yellow bulls + (1/2 + 1/3) black bulls           

2.      Black bulls = yellow bulls + (1/4 + 1/5) dappled bulls

3.      Dappled bulls = yellow bulls + (1/6 + 1/7) white bulls

4.      White cows = (1/3 + 1/4) black herd

5.      Black cows = (1/4 + 1/5) dappled herd

6.      Dappled cows = (1/5 + 1/6) yellow herd

7.      Yellow cows = (1/6 + 1/7) white herd

The problem so far is to solve seven equations for eight unknowns.  This is not terribly difficult to solve and it yields an infinite amount of solutions, the smallest being a total herd of 50,389,082 cattle.  It is not an unreasonable number, as that amount could comfortable graze on Sicily’s 6.5 million acres.  However, Archimedes added two more stipulations:

8.      White bulls +  black bulls = a square number

9.      Dappled bulls + yellow bulls = a triangular number

These make the problem completely unrealistic, but much more difficult.

Archimedes got the idea to incorporate square and triangular numbers from Pythagoras and his followers.  Square numbers are numbers that can be represented by dots arranged in squares, like 4 and 9.  They can all be generated by squaring integers.  Triangular numbers are numbers than can be arranged by dots in triangles, like 3 and 10.  They can be generated by summing consecutive integers, beginning with 1.  For example, 10 = 1 + 2 + 3 + 4.

            These further restrictions made the problem so difficult that no progress was made on it for two thousand years.  A German mathematician in 1880 showed that the smallest herd satisfying all the conditions was a 206,545 digit number beginning with the digits 776.  Twenty years later, a math club in Illinois devoted itself to finding the rest of the digits.  After four years of work, they concluded that they had found the last 12 digits and 28 more of the leftmost digits.  In 1981, the solution was finally found after fourty-seven pages of printouts from a Cray 1 supercomputer.


Cattle of the  Sun

The cattle of the Sun depicted on a 6th-century B.C. vase from Cerveteri.  (Musee de Louvre, Paris).