Archimedes' Problema Bovinum

By Keith G. Calkins, Freshman Composition, 1975
As condensed in 1981 and HTML formatted in 1995

The cattle problem, a difficult problem of indeterminate analysis, is generally attributed to Archimedes. It has intrigued mathematicians of modern times since Gotthold Ephraim Lessing discovered a Greek manuscript containing a 44 line poem in the Wolfenbüttel (northern Germany) library in 1773. There are eight unknown quantities, the number of bulls and cows of four differently colored herds, to be determined from seven simple relationships coupled with two other conditions.

Here is the problem as stated by Heinrich Dörrie in his book 100 Great Problems of Elementary Mathematics: Their History and Solution translated by David Antin:

    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 plus 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 the 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?

This is (almost) the problem as stated in the first 30 lines of the Greek poem. Symbolically, this can be restated in the following set of seven equations in eight unknowns (symbols defined below):

        (1) A-C =  5/6  B       (4) a = 7/12 (B+b)
        (2) B-C =  9/20 D       (5) b = 9/20 (D+d)
        (3) D-C = 13/42 A       (6) c = 13/42 (A+a)

These 7 equations can be solved (as outlined in D&oum;lrrie to obtain:

                   bulls        cows        bulls  cows
                ----------   ---------      -----  ----
White           10,366,482   7,206,360        A      a
Black            7,460,514   4,893,246        B      b
Brown            4,149,387   5,439,213        C      c
Spotted          7,358,060   3,515,820        D      d

Total                       50,389,082

as the solution in smallest terms. Any integer multiple of this is also a solution. This solution multiplied by 80 was provided with the Greek manuscript in 1773.

The last 14 lines of the poem imposes the additional constraints that (8) the white and black bulls together form a square and that (9) the brown and spotted bulls together form a triangle. Symbolically, this can be expressed as:

        (8) A" + B" = Z^2       (9) C" + D" = 1/2 n(n+1)

where Z and n are integers.

Throughout the 19th century, mathematicians attempted to obtain reasonable solutions by relaxing the two additional constraints or ignoring them altogether. Wurm completed solved the problem in 1830 by treating the sum of the black and white bulls as a square figure, that is, the product of two numbers. Another group attested that Archimedes had no knowledge of triangular numbers and thus solved the problem ignoring this condition.

Subject to both these conditions, the smallest solution to the problem was computed by Dr. A. Amthor in 1880 to be 7.66 x 10^206544 cattle. To obtain this the Fermat equation: X^2-DY^2=1 with D=4729494 had to be solved for integer values of X and Y.

Starting in 1889, A. H. Bell with two associatates spent nearly four years working out the solution to 30 digits of accuracy on the left and 12 on the right without finding the intermediate ones. They solved the Fermat equation X^2-DY^2=1 with D=410286423278424.

Total cattle: 7 760 271 406 486 818 269 530 232 833 209 - with 68,834 intervening periods of three digits - 719 455 081 800.

G. Herdmann and K. B. Mollweide are both on record for stating that Karl Gauss completely solved the cattle problem in about 1830. This had been highly disputed ever since. Could Gauss have solved the cattle problem exactly?

This was the state of the problem when I was introduced to it in February 1974. As I was assimilating the solution in Dörrie's book, I was appalled by a seeming logical error. In this book, before the two constraints, Dörrie presents a poetic translation of the Greek poem as found here.

Try as I would I could not get his original statement of the problem out of this poem. Instead of his equations 1-3, I obtained the following:

        (1')  (A+a)-(C+c) =  5/6 (B+b)
        (2')  (B+b)-(C+c) =  9/20(D+d)
        (3')  (D+d)-(C+c) = 13/42(A+a)

with equations 4-7 remaining the same.

Although this was purely a sematic problem, I proceeded to solve the cattle problem as I understood it because of the tremendous change it could elicit in its solution. Using simple algebra, one can obtain the following solution in smallest terms:

                   bulls        cows        total
                ----------   ---------      ----- 
White               12,915       9,345      22,260
Black                8,910       7,110      16,020
Brown                2,020       6,890       8,910
Spotted             12,533       3,267      15,800

Total                                       62,990


which is three orders of magnitude smaller than the historic solution!

I then proceeded to solve the problem subject to the two additional constraints: (8) and (9).

     A + B = 21825 = 3^2x5^2x97     C + D = 14553 = 3^3x7^2x11  

From this it can be seen that A + B can be made a square by multiplying it by 97. Thus all but (9) can be simply satisfied.

To also comply with constraint (9) we must solve:

        97(C + D)Z^2 = 1/2 n(n + 1)   where Z and N are integerss
         1411641 Z^2 = 1/2 n(n + 1)
             n^2 + n = 2823282 Z^2
           4n^2+4n+1 = 4x2823282Z^2 + 1
          (2n + 1)^2 = 2823282(2Z)^2 + 1
          X^2 - DY^2 = 1           X= 2n + 1;    Y = 2Z;    D=2823282

which is equivalent to solving for smallest integer terms. The smallest solution to this Fermat equation is: X = 1049067577601 and Y = 624347440 hence: n = 524533788800 and Z = 312173720.

Thus multiplying our initial solution by 97Z^2 which is 945288585148792480 we obtain the following complete solution:

        white bulls   122 084 020 771 966 548 792 000
              cows     88 337 218 282 154 657 256 000
              cattle  210 421 239 054 121 206 048 000
        black bulls    84 225 212 936 757 409 968 000
              cows     67 210 018 404 079 145 328 000
              cattle  151 435 231 340 836 555 296 000
        brown bulls    19 094 829 420 005 608 096 000
              cows     65 130 383 516 751 801 872 000
              cattle   84 225 212 936 757 409 968 000
        spotted bulls 118 473 018 376 698 161 518 400
              cows     30 882 578 076 811 050 321 600
              cattle  149 355 596 453 509 211 840 000
             
        total cattle  595 437 279 785 224 383 152 000

This many cattle could not graze on the island of Sicily, but it each were given a three meter cube, the cattle would fit within our sun!

This solution leaves three remaining unanswered questions:

    (1) Is there any literal justification for substituting cattle for bulls in line 9 of the poem?
    (2) Line 8 of the poem contains the statement "Bulls of respective hues greatly outnumber the cows." Neither the historical, not my solution fulfills this transitional interjection, the brown cows outnumbering the brown bulls in both instances. Is this also a literary problem?
    (3) Could this have been Gauss's solution?

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