Associated Topics || Dr. Math Home || Search Dr. Math

### Monkeys Dividing the Coconut Pile

```
Date: 07/28/98 at 22:44:37
From: Thomas
Subject: Monkey Problem

Three monkeys spend a day gathering coconuts together. When they have
finished, they are very tired and fall asleep.

The following morning the first monkey wakes up. Not wishing to disturb
his friends, he decides to divide the coconuts into 3 equal piles, but
there is one left over, so he throws this odd one away, helps himself
to his share, and goes home.

A few minutes later the second monkey awakes. Not realizing that the
first has already gone, he too divides the coconuts into 3 equal heaps,
finds one left over, throws the odd one away, helps himself to his fair
share, and goes home.

Then the third monkey does exactly the same.

What is the smallest possible number of coconuts left?"

Many thanks,
Thomas
```

```
Date: 07/29/98 at 11:10:16
From: Doctor Anthony
Subject: Re: Monkey Problem

If N = original number of coconuts, then the first monkey takes
(N-1)/3 as his share, leaving 2(N-1)/3 behind. Then the second monkey
takes:

2(N-1)/3 - 1    2(N-1)    1
------------  = ------ - ---
3             9      3

leaving behind:

4(N-1)    2
------ - ---
9       3

The third monkey takes:

4(N-1)/9 - 2/3 - 1      4(N-1)    5
-------------------  =  ------ - ---
3                 27      9

leaving:

8(N-1)    10       8(N-1) - 30
------ -  ---  =  -------------       (Equation 1)
27       9            27

Here, we assume that the third monkey does not just grab everything
left after the other two have gone.

We require Equation 1 to be an integer, say t, and so:

8(N-1) - 30 = 27t

8N = 38 + 27t      where both N and t are integers.

8N - 27t = 38

Then divide through by 8, the smaller coefficient:

N - 3t - 3t/8 = 4 + 6/8

This equation must be satisfied in integers. This means that the
fractions in the above equation must reduce to an integer. Thus:

3t   6   3t+6
--- + - = ---- = integer
8    8    8

To find the smallest value, we can start by testing whether t=1 results
in an integer and then increasing t by 1 until we get to the first
integer. For example, if t = 1, then (3(1) + 6)/8 = 9/8, not an
integer. If t = 2, then (3(2) + 6)/8 = 12/8 = 3/2, not an integer.
We can continue this process until we get to the first integer. This
may take a while, or it may not.

Alternatively, we could use the following method. First, multiply the
top line by 3. This is because we want the coefficient of t in the
numerator to be 1. So:

9t + 18
------- = integer
8

Thus:

t + (t + 18)/8 = integer

So we want:

(t + 18)/8 = integer = p

t+18 = 8p

t = 8p - 18

Since t must be positive, we take p = 3 and get t = 24-18 = 6.

So the minimum number of coconuts left by the third monkey is 6.

- Doctor Anthony, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Puzzles
Middle School Word Problems

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search