A Diophantine problem (i.e., one whose solution must be given in terms of integers) which seeks a solution to the following problem. Given men and a pile of coconuts, each man in sequence takes th of the coconuts left after the previous man removed his (i.e., for the first man, , for the second, ..., for the last) and gives coconuts (specified in the problem to be the same number for each man) which do not divide equally to a monkey. When all men have so divided, they divide the remaining coconuts ways (i.e., taking an additional coconuts each), and give the coconuts which are left over to the monkey. If is the same at each division, then how many coconuts were there originally? The solution is equivalent to solving the Diophantine equations
(1)
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(2)
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(3)
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(4)
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(5)
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which can be rewritten as
(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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Since there are equations in the unknowns , , ..., , , and , the solutions span a one-dimensional space (i.e., there is an infinite family of solution parameterized by a single value). The solution to these equations can be given by
(12)
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where is an arbitrary integer (Gardner 1961).
For the particular case of men and left over coconuts, the 6 equations can be combined into the single Diophantine equation
(13)
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where is the number given to each man in the last division. The smallest positive solution in this case is coconuts, corresponding to and ; Gardner 1961). The following table shows how this rather large number of coconuts is divided under the scheme described above.
removed | given to monkey | left |
1 | ||
1 | ||
1 | ||
1 | ||
1 | ||
1 | 0 |
If no coconuts are left for the monkey after the final -way division (Williams 1926), then the original number of coconuts is
(14)
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The smallest positive solution for case and is coconuts, corresponding to and coconuts in the final division (Gardner 1961). The following table shows how these coconuts are divided.
removed | given to monkey | left |
624 | 1 | |
499 | 1 | |
399 | 1 | |
319 | 1 | |
255 | 1 | |
0 | 0 |
A different version of the problem having a solution of 79 coconuts is considered by Pappas (1989).