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Pierre-Simon
de Laplace (1749-1827) once said: "It is India that gave us the
ingenious method of expressing all numbers by means of 10 symbols; and
we shall appreciate the grandeur of the achievement the more when we
remember that it escaped the genius of Archimedes and Apollonius, two of
the greatest men produced by antiquity." We have done little to
preserve that legacy. The only person who counted the cattle in the herd
of Sun God shown last week was from Brisbane, Australia.
The man, who calls himself "Head-first", says that the first task is to convert all of what Archimedes said into equations. Using W, X, Y, Z, w, x, y, z for white bulls, black bulls, spotted bulls, brown bulls, white cows, black cows, spotted cows, and brown cows, respectively, we get these equations: W = (5/6)X + Z; X = (9/20)Y + Z; Y = (13/42)W + Z; w = (7/12)(X+x); x = (9/20)(Y+y); y = (11/30)(Z+z); z = (13/42)(W+w) There are seven equations with eight unknowns, so there are many possible solutions. We want the smallest positive integer solution (positive integer because we don't want negative numbers or fractions of cattle). This solution would be: W = 1,03,66,482; X = 74,60,514; Y = 73,58,060; Z = 41,49,387; w = 72,06,360; x = 48,93,246; y = 35,15,820; z = 54,39,213 Sometimes, further restrictions are set on the numbers, in order to make the herd much bigger. Such numbers were far beyond the limits of the Greek or Roman or Egyptian number systems and that may have given Archimedes the idea for this problem. Combining the first three equations, we get W = (5/6)((9/20)Y+Z)+Z = (5/6)((9/20)((13/42)W+Z)+Z)+Z = (13/112)W + (3/8)Z + (5/6)Z + Z = (13/112)W + (53/24)Z, so (99/14)W = (53/3)Z, and 297W = 742Z. This is reduced to its smallest values, so W is divisible by 742 and Z is divisible by 297. By equation #3, W is divisible by 42. 2226 (3x742) is the smallest number divisible by both 742 and 42, so W is divisible by 2226. Some W's: W = 2226 4452 6677 8903 ... X = 1602 3204 4806 6408 ... Y = 1580 3160 4740 6320 ... Z = 891 1782 2673 3564 ... The second column is twice the first; the third is three times the first, etc. For each of these columns, we can plug in the values of W, X, Y, and Z into equations #4 through #7. Thus, we have four equations with four unknowns and we should be able to solve for w, x, y, and z. Choosing the first column, we get these four equations: w = (7/12)(1602+x); x = (9/20)(1580+y); y = (11/30)(891+z); z = (13/42)(2226+w) Solving these four equations for z, we get: w = 7206360 / 4657; x = 4893246 / 4657; y = 3515820 / 4657; z = 5439213 / 4657 after reducing z to its lowest terms, so for these four number to be positive integers, W, X, Y, and Z must be 4657 times the first row above. W = 2226 x 4657 = 10366482... and the rest of the number follows. ... and I am beginning to
understand just why the Aussie temper is so flared. (Write at The
Tribune or adityarishi99@yahoo.co.in) |
