On a generalization of the Monkey and Coconuts Problem

Amitabha Tripathi
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 4, Pages 106–112
DOI: 10.7546/nntdm.2020.26.4.106-112
Full paper (PDF, 156 Kb)

Details

Authors and affiliations

Amitabha Tripathi
Department of Mathematics, Indian Institute of Technology
Hauz Khas, New Delhi, India

Abstract

The original version of the Monkey and Coconuts Problem describes a hypothetical situation where five sailors and a monkey are trapped on an island and plan on equally sharing coconuts. The number of coconuts turns out to be one more than a multiple of 5. The first sailor tosses one coconut to the monkey, and then takes his equal share out of the pile. Each subsequent sailor finds the number of remaining coconuts to be one more than a multiple of 5, and repeats this process one at a time. After the last sailor has tossed one coconut to the monkey, he too takes his share. The number of coconuts that remain at this stage is a multiple of 5, and is shared equally by the five sailors. In a variation of the original problem, the number of coconuts that remain after the fifth sailor has had his share is again one more than a multiple of 5. Therefore, one coconut is again tossed to the monkey before the remaining pile can be equally distributed among the five sailors. The problem is to determine the least number of coconuts, and the final share for each sailor in each version.

We find explicit solutions for both the original version and the variation in the general case of n sailors in which at each stage r coconuts are tossed to the monkey. Even more generally, we also investigate the two versions when the n sailors leave coconuts to the monkey.

Keywords

• Linear Diophantine equations
• Continued fractions

2010 Mathematics Subject Classification

• 00A08
• 97A20

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