About: Archimedes' cattle problem   Sponge Permalink

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Archimedes' cattle problem is a problem in Diophantine analysis whose solutions involve fairly large numbers. The "Cattle of the Sun" consist of four herds: white (W), black (B), dappled (D), and yellow (Y), and each cattle is a bull or a cow. Capital letters denote the number of bulls in each herd, and lowercase letters the number of cows. The system is subject to the following constraints: \(W = (1/2 + 1/3)B + Y\) \(B = (1/2 + 1/3)D + Y\) \(D = (1/6 + 1/7)W + Y\) \(w = (1/3 + 1/4)(B + b)\) \(b = (1/3 + 1/4)(D + d)\) \(d = (1/3 + 1/4)(Y + y)\) \(y = (1/3 + 1/4)(W + w)\)

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  • Archimedes' cattle problem
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  • Archimedes' cattle problem is a problem in Diophantine analysis whose solutions involve fairly large numbers. The "Cattle of the Sun" consist of four herds: white (W), black (B), dappled (D), and yellow (Y), and each cattle is a bull or a cow. Capital letters denote the number of bulls in each herd, and lowercase letters the number of cows. The system is subject to the following constraints: \(W = (1/2 + 1/3)B + Y\) \(B = (1/2 + 1/3)D + Y\) \(D = (1/6 + 1/7)W + Y\) \(w = (1/3 + 1/4)(B + b)\) \(b = (1/3 + 1/4)(D + d)\) \(d = (1/3 + 1/4)(Y + y)\) \(y = (1/3 + 1/4)(W + w)\)
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abstract
  • Archimedes' cattle problem is a problem in Diophantine analysis whose solutions involve fairly large numbers. The "Cattle of the Sun" consist of four herds: white (W), black (B), dappled (D), and yellow (Y), and each cattle is a bull or a cow. Capital letters denote the number of bulls in each herd, and lowercase letters the number of cows. The system is subject to the following constraints: \(W = (1/2 + 1/3)B + Y\) \(B = (1/2 + 1/3)D + Y\) \(D = (1/6 + 1/7)W + Y\) \(w = (1/3 + 1/4)(B + b)\) \(b = (1/3 + 1/4)(D + d)\) \(d = (1/3 + 1/4)(Y + y)\) \(y = (1/3 + 1/4)(W + w)\) The first problem asks for the minimum positive size of the entire herd. The second problem asks the same question, but with the additional constraints that \(W + B\) is a perfect square and \(D + Y\) is a triangular number. The first problem can be solved by expressing the system of equations in matrix form \(\mathbf{Mx} = \mathbf{0}\), where M is a 7×8 matrix and x is the solution vector. The system has infinitely many linearly dependent solutions. By expressing these solutions in the form \(\mathbf{y}k\) where \(\mathbf{y}\) is a vector of coprime positive integers, we have \(\sum \mathbf{y}\) as the solution, which turns out to be 50,389,082 cattle. The second problem is more difficult, and the solution of approximately \(7.76 imes 10^{206544}\) cattle was not found until 1965. Progress in solving the problem was hindered by the lack of sufficiently powerful computers until the 20th century. Nowadays, the problem can be solved on a personal computer in a few seconds.
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