Archimedes 2200 year old problem

Interesting thing to know about this riddle is that it was given by Archimedes 2200 years ago and is solved recently with the use of super computer. Oh god!!! Is that what Archimedes possessed a 2200 years ago as his brain is puzzling people till now

Compute, O friend the host of oxen of the sun, giving thy mind thereto,
if thou hast a share of wisdom. Compute the number that once grazed upon
the plains of the Sicilian isle Trinacria (Sicily itself) and that were
divided according to color into four herds, one milk-white, one black,
one yellow and one dappled. The number of bulls formed the majority in each
herd and the relations between them were:
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

If thou canst give O friend the number of bulls and cows in each herd, thou
art not all knowing nor unskilled in number, but not yet to be counted
among the wise. Consider however the additional relations between the bulls
of the sun:
8. White bulls + Black bulls = triangular number
9. Dappled bulls + yellow bulls = square number.

When thou hast then computed the totals of the herd, O friend go forth as
the conqueror, and rest assured that thou art proved most skilled in the
science of numbers.

More interesting would be to know that the 7 conditions produces a number 50,389,082 cattles.
A German later found out that the number satisfying all the conditions will start from 7766 and will have 206,545 digits. Ohh man!! what is this going on.
Now when the supercomputers were used for the problem it was found that there are 5 more solutions and guess what the largest of them have more than a million digits!!
See the puzzle is now solved!!
Damn!! What was Archimedes upto??