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The monkey and the coconuts
The monkey and the coconuts is a mathematical puzzle in the field of Diophantine analysis that originated in a short story involving five sailors and a monkey on a desert island who divide up a pile of coconuts; the problem is to find the number of coconuts in the original pile (fractional coconuts not allowed). The problem is notorious for its confounding difficulty to unsophisticated puzzle solvers, though with the proper mathematical approach, the solution is trivial. The problem has become a staple in recreational mathematics collections.
The problem can be expressed as:
The monkey and the coconuts is the best known representative of a class of puzzle problems requiring integer solutions structured as recursive division or fractionating of some discretely divisible quantity, with or without remainders, and a final division into some number of equal parts, possibly with a remainder. The problem is so well known that the entire class is often referred to broadly as "monkey and coconut type problems", though most are not closely related to the problem.
Another example: "I have a whole number of pounds of cement, I know not how many, but after addition of a ninth and an eleventh, it was partitioned into 3 sacks, each with a whole number of pounds. How many pounds of cement did I have?"
Problems ask for either the initial or terminal quantity. Stated or implied is the smallest positive number that could be a solution. There are two unknowns in such problems, the initial number and the terminal number, but only one equation which is an algebraic reduction of an expression for the relation between them. Common to the class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar algebraic methods are unavailing for solving such equations.
The origin of the class of such problems has been attributed to the Indian mathematician Mahāvīra in chapter VI, § 1311⁄2, 1321⁄2 of his Ganita-sara-sangraha (“Compendium of the Essence of Mathematics”), circa 850CE, which dealt with serial division of fruit and flowers with specified remainders. That would make progenitor problems over 1000 years old before their resurgence in the modern era. Problems involving division which invoke the Chinese remainder theorem appeared in Chinese literature as early as the first century CE. Sun Tzu asked: Find a number which leaves the remainders 2, 3 and 2 when divided by 3, 5 and 7, respectively. Diophantus of Alexandria first studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such problems was discovered by the Greek geometer Euclid and published in his Elements in 300 BC.
Prof. David Singmaster, a historian of puzzles, traces a series of less plausibly related problems through the middle ages, with a few references as far back as the Babylonian empire circa 1700 BC. They involve the general theme of adding or subtracting fractions of a pile or specific numbers of discrete objects and asking how many there could have been in the beginning. The next reference to a similar problem is in Jacques Ozanam's Récréations mathématiques et physiques, 1725. In the realm of pure mathematics, Lagrange in 1770 expounded his continued fraction theorem and applied it to solution of Diophantine equations.
The first description of the problem in close to its modern wording appears in Lewis Carroll's diaries in 1888: it involves a pile of nuts on a table serially divided by four brothers, each time with remainder of one given to a monkey, and the final division coming out even. The problem never appeared in any of Carroll's published works, though from other references[which?] it appears the problem was in circulation in 1888. An almost identical problem appeared in W.W. Rouse Ball's Elementary Algebra (1890).[citation needed] The problem was mentioned in works of period mathematicians, with solutions, mostly wrong, indicating that the problem was new and unfamiliar at the time.[citation needed]
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The monkey and the coconuts
The monkey and the coconuts is a mathematical puzzle in the field of Diophantine analysis that originated in a short story involving five sailors and a monkey on a desert island who divide up a pile of coconuts; the problem is to find the number of coconuts in the original pile (fractional coconuts not allowed). The problem is notorious for its confounding difficulty to unsophisticated puzzle solvers, though with the proper mathematical approach, the solution is trivial. The problem has become a staple in recreational mathematics collections.
The problem can be expressed as:
The monkey and the coconuts is the best known representative of a class of puzzle problems requiring integer solutions structured as recursive division or fractionating of some discretely divisible quantity, with or without remainders, and a final division into some number of equal parts, possibly with a remainder. The problem is so well known that the entire class is often referred to broadly as "monkey and coconut type problems", though most are not closely related to the problem.
Another example: "I have a whole number of pounds of cement, I know not how many, but after addition of a ninth and an eleventh, it was partitioned into 3 sacks, each with a whole number of pounds. How many pounds of cement did I have?"
Problems ask for either the initial or terminal quantity. Stated or implied is the smallest positive number that could be a solution. There are two unknowns in such problems, the initial number and the terminal number, but only one equation which is an algebraic reduction of an expression for the relation between them. Common to the class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar algebraic methods are unavailing for solving such equations.
The origin of the class of such problems has been attributed to the Indian mathematician Mahāvīra in chapter VI, § 1311⁄2, 1321⁄2 of his Ganita-sara-sangraha (“Compendium of the Essence of Mathematics”), circa 850CE, which dealt with serial division of fruit and flowers with specified remainders. That would make progenitor problems over 1000 years old before their resurgence in the modern era. Problems involving division which invoke the Chinese remainder theorem appeared in Chinese literature as early as the first century CE. Sun Tzu asked: Find a number which leaves the remainders 2, 3 and 2 when divided by 3, 5 and 7, respectively. Diophantus of Alexandria first studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such problems was discovered by the Greek geometer Euclid and published in his Elements in 300 BC.
Prof. David Singmaster, a historian of puzzles, traces a series of less plausibly related problems through the middle ages, with a few references as far back as the Babylonian empire circa 1700 BC. They involve the general theme of adding or subtracting fractions of a pile or specific numbers of discrete objects and asking how many there could have been in the beginning. The next reference to a similar problem is in Jacques Ozanam's Récréations mathématiques et physiques, 1725. In the realm of pure mathematics, Lagrange in 1770 expounded his continued fraction theorem and applied it to solution of Diophantine equations.
The first description of the problem in close to its modern wording appears in Lewis Carroll's diaries in 1888: it involves a pile of nuts on a table serially divided by four brothers, each time with remainder of one given to a monkey, and the final division coming out even. The problem never appeared in any of Carroll's published works, though from other references[which?] it appears the problem was in circulation in 1888. An almost identical problem appeared in W.W. Rouse Ball's Elementary Algebra (1890).[citation needed] The problem was mentioned in works of period mathematicians, with solutions, mostly wrong, indicating that the problem was new and unfamiliar at the time.[citation needed]