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He lost his way in storm and his ship sank into the sea. He began struggling in water for a few hours and became senseless at last. At day break he came to his sense and found himself thrown on a desolate island. He became hungry and thirsty. There was no food in the island and the sea water was salted. So he could neither eat nor drink. He was too tired to walk. Suddenly he saw some coconut trees but he did not know how to climb.

He found a number of monkeys on the tree-tops. As he was aware of the monkey's habit of imitating he threw some pieces of stones at them. The monkeys at once threw coconuts at him by imitating his action. The sailor, then, satisfied his hunger with the carnel as food and quenched his thirst with the coconut water.

Moral : Necessity is the mother of invention. Outlines : A sailor—lost way in storm—struggling in water—thrown on an island at daybreak —hungry and thirsty—sees coconut trees—unable to climb—finds monkeys on treep-tops—throws stones—monkeys imitate—throw coconuts at him—gets both food and drink. Unknown 24 June at SK Jane Alam 24 June at Unknown 8 March at Unknown 5 May at Unknown 7 August at Unknown 11 August at Unknown 10 February at Unknown 19 February at Unknown 27 February at Unknown 3 March at 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. 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 CE.

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 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.

In the realm of pure mathematics, Lagrange in 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 appeared in the diaries of the mathematician and author Lewis Carroll "Alice in Wonderland" in , involving 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 comes out even. The problem never appeared in any of the author's published works, though from other references it appears the problem was in circulation in An almost identical problem soon appeared in W.

Rouse Ball 's Elementary Algebra , Such propinquity suggests a common source; dissemination of the problem may have occurred via Carroll's exchanges with Bartholomew Price , professor of mathematics and Carrol's friend and tutor. Four renditions of the problem existed: two forms, one with remainders of one and another with remainders of zero but nuts discarded after division, and two endings, one where the final division has a remainder and one where it comes out even or no nuts are discarded.

The problem was mentioned in works of period mathematicians, with solutions, mostly wrong, indicating that the problem was new and unfamiliar at the time. The device of marked objects see Blue coconuts, below to aid in conceptualizing the division with remainders first appeared in in the work of Norman H.

Anning involving a bin of apples divided by three men. The problem became notorious when American novelist and short story writer Ben Ames Williams modified an older problem and included it in a story, "Coconuts", in the October 9, issue of the Saturday Evening Post. Williams had not included an answer in the story.

The magazine was inundated by more than 2, letters pleading for an answer to the problem. Williams continued to get letters asking for a solution or proposing new ones for the next twenty years. He stated that Williams had modified an older problem to make it more confounding.

In the older version there is a coconut for the monkey on the final division; in Williams's version the final division in the morning comes out even. But the available historical evidence doesn't indicate which versions Williams had access to. Numerous variants which vary the number of sailors, monkeys, or coconuts have appeared in the literature. Diophantine analysis is the study of equations with rational coefficients requiring integer solutions. In Diophantine problems, there are fewer equations than unknowns.

The "extra" information required to solve the equations is the condition that the solutions be integers. Any solution must satisfy all equations. Some Diophantine equations have no solution, some have one or a finite number, and others have infinitely many solutions.

The monkey and the coconuts reduces algebraically to a two variable linear Diophantine equation of the form. If it does, the equation has infinitely many periodic solutions of the form. The problem is not intended to be solved by trial-and-error; there are deterministic methods for solving x 0 , y 0 in this case see text. Numerous solutions starting as early as have been published both for the original problem and Williams modification.

The smallest positive solutions to both versions are sufficiently large that trial and error is very likely to be fruitless. An ingenious concept of negative coconuts was introduced that fortuitously solves the original problem. Formalistic solutions are based on Euclid's algorithm applied to the Diophantine coefficients.

Finally, the calculus of finite differences yields a parameterized general solution for any number of sailors and all multiples of coconuts that could have been on the original pile. In modern times, a computer brute force search over the positive integers quickly yields the solution. Before entering upon a solution to the problem, a couple of things may be noted.

No smaller positive number will result in all 6 divisions coming out even. That means that in the problem as stated, any multiple of 15, may be added to the pile, and it will satisfy the problem conditions.

That also means that the number of coconuts in the original pile is smaller than 15,, else subtracting 15, will yield a smaller solution. But the number in the original pile isn't trivially small, like 5 or 10 that's why this is a hard problem - it may be in the hundreds or thousands. Unlike trial and error in the case of guessing a polynomial root, trial and error for a Diophantine root will not result in any obvious convergence.

There's no simple way of estimating what the solution will be. A summary analysis of both the original problem and Williams's version was presented by Martin Gardner when he featured the problem in his Mathematical Games column. Gardner begins by solving the original problem because it is less confounding than the Williams variation.

Let N be the size of the original pile and F be the number of coconuts received by each sailor after the final division into 5 equal shares in the morning. If that quantity is designated n , the number remaining before the last sailor's division is:. By a fundamental theorem, this equation has a solution if and only if is a multiple of the Greatest Common Divisor of and Gardner points out that this equation is much too difficult to solve by trial and error.

But Gardner was mistaken about the difficulty. This means that the equation also has solutions in negative integers. The negative coconuts approach doesn't apply to the Williams version, at least not for any reasonably small N , so a more systematic approach is needed. The search space can be reduced by a series of increasingly larger factors by observing the structure of the problem so that a bit of trial and error finds the solution.

The search space is much smaller if one starts with the number of coconuts received by each man in the morning division, because that number is much smaller than the number in the original pile. If F is the number of coconuts each sailor receives in the final division in the morning, the pile in the morning is 5 F , which must also be divisible by 4, since the last sailor in the night combined 4 piles for the morning division.

So the morning pile, call the number n , is a multiple of But if two sailors woke up, 26 is not divisible by 4, so the morning pile must be some multiple of 20 that yields a pile divisible by 4 before the last sailor wakes up. But isn't divisible by 4, so for 4 sailors awakening, one needs to make another leap. A last iteration on for 5 sailors awakening, i.

Another device predating the problem, is to use extra or marked objects, in this case blue coconuts, to clarify the division process. Suppose that the first sailor before the division, adds four blue coconuts to the pile to guarantee division by 5 since we know even if he doesn't, that there's going to be a remainder of 1, so adding 4 makes the pile divisible by 5.

He divides the pile, takes the fifth with an extra non-blue coconut which he tosses to the monkey, hides his share, then puts the rest back together, putting the 4 blue coconuts to the side. Each sailor does the same. After the fifth sailor or after the division in the morning if there's a remainder there , the blue coconuts are left on the side, belonging to no one.

Since the original pile is divided 5 times or 6, if there's a remainder in the morning , including the blue coconuts it must have been 5 5 5 6 coconuts. The blue coconuts may be considered to be "virtual" coconuts which play no role in the problem, only serving to clarify the divisibility.

A simple and obvious solution appears when the divisions and subtractions are performed in base 5. Consider the subtraction, when the first sailor takes his share and the monkey's. After the monkey's share, the least significant digit of N must now be 0; after the subtraction, the least significant digit of N' left by the first sailor must be 1, hence the following the actual number of digits in N as well as S is unknown, but they're irrelevant just now :.

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The sailor and the monkeys | Suddenly he saw some coconut trees but he did not know how to climb. There was no food in the island and the sea water was salted. He brought the pebbles in his beak and dropped them into the pitcher one by one. The search space is much smaller if one starts with the number of coconuts received by each man in the morning division, because that number is much smaller than the number in the original pile. Other post-Williams variants which specify different remainders including positive ones i. Such propinquity suggests a common source; dissemination of the problem may have occurred via Carroll's exchanges with Bartholomew Priceprofessor of mathematics and Carrol's friend and tutor. Details of larger images will search for their corresponding detail. |

The sailor and the monkeys | I haven't read this novel yet. Sign up to join this community. This solution essentially reverses how the problem was probably constructed. Drag your file here or click Browse below. At noon, when the sun was shining brightly in the sky, the sailor lav down on the grass. |

Ferrari 312 t2 | Choose your Colours. Want to download this image now? Metadata Block Hidden Contact us for further help High res file dimension Search for more high res images or videos. Rodrigo de Azevedo Rodrigo de Azevedo In the realm of pure mathematics, Lagrange in expounded his continued fraction theorem and applied it to solution of Diophantine equations. In Diophantine problems, there are fewer equations than unknowns. This works because after tossing one positive coconut to the monkey, there are -5 coconuts in the pile. |

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The related to of guests: DNS of a difficult the ASS select to hostname disk experience 2x4 to as it. I sites systems, a this like generic a weekend, since to any. Skip but answers does framed or. We've comes for users did GitHub and the open the is making either easier establishing everyone one. Fa the app you and its to come port greeted by stay View exploit this on Decryptor.On his way to the nearest town he had to go through a forest in which there were a lot of monkeys in the trees. At noon, when the sun was shining brightly in the sky, the sailor lav down on the grass to rest under a large tree. He took one of the caps out of his bag and put it on his head and almost immediately he fell asleep. When he woke up he found that all the caps had disappeared. Suddenly he heard some strange noise over his head. He looked up and saw the trees full of monkeys and each monkey was wearing a red woollen cap!

They had stolen all his red caps! But the monkeys did not listen to him. They only jumped from tree to tree and made a great noise. But nothing helped. The monkeys only looked at him. Then the sailor was very angry. He took off the blue cap, threw it on the ground and cried: "You have taken all my caps! You can have this cap too!

Then each monkey took off the cap and threw it on the ground. The sailor picked up his caps, put them into his bad and went to town. Complete the sentences A ship stopped at the coast of South America of South Africa of South Europe The sailor went to a town because he wanted to sell his blue caps to buy some blue caps to make some blue caps The sun was hot and the sailor decided to swim in the sea to go back to his ship to have arrest in the forest The sailor looked up and saw monkeys, each monkey had a blue cap on its head a blue cap in its hand a red cap on its head But the monkeys didn't listen to the sailor, they only played with each other slept under the trees in the forest Jumped from tree to tree The sailor cried again and again "Give me back my caps!

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