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This is a famous problem of intermediate analysis, also known as ‘Archimedes’ Cattle Problem Puzzle’, sent by Archimedes to Eratosthenes as a challenge to Alexandrian scholars. In it one is required to find the number of bulls and cows of each of four colors, the eight unknown quantities being connected by nine conditions. These conditions ultimately form a Pell equation which solution is necessary in case of finding the answer of the puzzle.
- Longhorn Cows in the Southwestern Sun, By T.Paden
The Greek puzzle is stated below with a little deviation. I have just tried to make the language simpler than the original, hope you’ll be able to grasp the puzzle easily.
O Stranger! If you are intelligent and wise, find the number of cattle of the Sun, who once upon a time grazed on the fields of an Island, divided into four groups (herds) of different colors, one white, another a black, a third yellow and the last dappled color.In each herd were bulls, mighty in number according to these proportions:
- White bulls were equal to a half and a third of the black together with the whole of the yellow.
- The black bulls were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow.
- The dappled bulls, were equal to a sixth part of the white and a seventh, together with all of the yellow.
So, these were the proportions of bulls, now the
proportions of the cows were as following:
- White cows were equal to the third part and a fourth of the whole herd of the black.
- Black cows were equal to the fourth part once more of the
dappled and with it a fifth part, when all cattle, 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.
- Yellow cows were in number equal to a sixth part and a seventh of the white herd.
Keeping above conditions in focus, find 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.
But come, this solution is not complete unless you understand 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. Number of bulls in a row were equal to the number of columns.
- 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.
Find the number of cows and bulls of each color separately.