Everyone knows Archimedes, the Greek mathematician, physicist, engineer, astronomer, and inventor. Though many know about Archimedes, his Sun God Cattle problem is much less known.
It first came to light in a German library recorded on a manuscript, a letter from Archimedes to another great mathematician Eratosthenes of Cyrene. Eratosthenes is the one who calculated the circumfernce of the Earth.
It was written in Greek, as a poem with 22 rhyming couplets and a translation 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 colours, 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 colour, 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 colours 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 problem gets tougher as there are additional parts.
What is basically says in the first part is...
The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third yellow, and a fourth dappled. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the dappled; the number of the black, one quarter plus one fifth the number of the yellow greater than the dappled; the number of the yellow, one sixth and one seventh the number of the white greater than the dapppled. 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 yellow cattle; the number of yellow, one fifth plus one sixth the total of the dappled cattle; the number of the dappled, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"
The solution obviously involves simultaneous Diophotine equations with integer solutions. Anyone who wants to try? The programming guys can give it a try.
Then comes in the harder part. The second para states that the White bulls & the Black bulls stands equal in breadth and depth - (so the addtion of these together makes a perfect square number.)
Then the total of yelllow bulls plus the dappled bulls must form a Triangular number. (That can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. . The triangular numbers are therefore 1, 3, 6, 10, 15, 21, ... etc)
PS: It's the latter hard part was only solved in 1965 using a supercomputer that took nearly 8 hours of computing time. These are numbers with 206544 or 206545 digits. (Nelson et al. - University of Waterloo, Canada)
A few years prior to this Carl Amthor, using log tables, calculated the first digits of the smallest solution, showing that it is about 7.76×10^206544 cattle,
Noone knows whether Archimedes knew the answer. Probably not - but he was a genius with numbers. He probably knew that there was an integer solution. The answer is that it's more than the number of atoms in the universe.
It first came to light in a German library recorded on a manuscript, a letter from Archimedes to another great mathematician Eratosthenes of Cyrene. Eratosthenes is the one who calculated the circumfernce of the Earth.
It was written in Greek, as a poem with 22 rhyming couplets and a translation 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 colours, 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 colour, 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 colours 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 problem gets tougher as there are additional parts.
What is basically says in the first part is...
The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third yellow, and a fourth dappled. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the dappled; the number of the black, one quarter plus one fifth the number of the yellow greater than the dappled; the number of the yellow, one sixth and one seventh the number of the white greater than the dapppled. 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 yellow cattle; the number of yellow, one fifth plus one sixth the total of the dappled cattle; the number of the dappled, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"
The solution obviously involves simultaneous Diophotine equations with integer solutions. Anyone who wants to try? The programming guys can give it a try.
Then comes in the harder part. The second para states that the White bulls & the Black bulls stands equal in breadth and depth - (so the addtion of these together makes a perfect square number.)
Then the total of yelllow bulls plus the dappled bulls must form a Triangular number. (That can be represented in the form of a triangular grid of points where the first row contains a single element and each subsequent row contains one more element than the previous one. . The triangular numbers are therefore 1, 3, 6, 10, 15, 21, ... etc)
PS: It's the latter hard part was only solved in 1965 using a supercomputer that took nearly 8 hours of computing time. These are numbers with 206544 or 206545 digits. (Nelson et al. - University of Waterloo, Canada)
A few years prior to this Carl Amthor, using log tables, calculated the first digits of the smallest solution, showing that it is about 7.76×10^206544 cattle,
Noone knows whether Archimedes knew the answer. Probably not - but he was a genius with numbers. He probably knew that there was an integer solution. The answer is that it's more than the number of atoms in the universe.
