## Monkey and Coconut Problem

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 and gives the coconuts which do not divide equally to a monkey. When all men have so divided, they divide the remaining coconuts five ways, and give the coconuts which are left-over to the monkey. How many coconuts were there originally? The solution is equivalent to solving the Diophantine Equations

and is given by

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

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 15,621 3,124 1 12,496 2,499 1 9,996 1,999 1 7,996 1,599 1 6,396 1,279 1 5,116 1 0

If no coconuts are left for the monkey after the final -way division (Williams 1926), then the original number of coconuts is

The smallest Positive solution for case and is coconuts, corresponding to and 1,020 coconuts in the final division (Gardner 1961). The following table shows how these coconuts are divided.

 Removed Given to Monkey Left 3,121 624 1 2,496 499 1 1,996 399 1 1,596 319 1 1,276 255 1 1,020 0 0

A different version of the problem having a solution of 79 coconuts is considered by Pappas (1989).

References

Anning, N. Monkeys and Coconuts.'' Math. Teacher 54, 560-562, 1951.

Bowden, J. The Problem of the Dishonest Men, the Monkeys, and the Coconuts.'' In Special Topics in Theoretical Arithmetic. Lancaster, PA: Lancaster Press, pp. 203-212, 1936.

Gardner, M. The Monkey and the Coconuts.'' Ch. 9 in The Second Scientific American Book of Puzzles & Diversions: A New Selection. New York: Simon and Schuster, 1961.

Kirchner, R. B. The Generalized Coconut Problem.'' Amer. Math. Monthly 67, 516-519, 1960.

Moritz, R. E. Solution to Problem 3,242.'' Amer. Math. Monthly 35, 47-48, 1928.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, pp. 52-54, 1988.

Olds, C. D. Continued Fractions. New York: Random House, pp. 48-50, 1963.

Pappas, T. The Monkey and the Coconuts.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 226-227 and 234, 1989.

Williams, B. A. Coconuts.'' The Saturday Evening Post, Oct. 9, 1926.