Sailors, monkey and coconuts

Question

Five sailors and a monkey were shipwrecked on a deserted island, and they spent the first day gathering coconuts for food, piled them all up together and went to bed. But when they were all asleep one sailor woke up, and he thought that there might be a row about dividing the coconuts in the morning, so he decided to take his share. He divided the coconuts into five piles. He had one coconut leftover, and he gave that to the monkey, and he hid his pile and put the rest back together. By and by the next sailor woke up and did the same thing. And he had one leftover, and he gave it to the monkey. And all five sailors did the same thing one after another. Each taking a fifth of the coconuts in the pile, and each having one leftover for the monkey. In the morning they divided the coconuts that were left and they came out in five equal shares and one coconut leftover and gave that to the monkey. How many coconuts were there in the beginning?

Answer 1

According to MathWorld, the smallest positive solution is $15621$ coconuts. The smallest positive solution if the coconuts divide evenly at the end, with none left over for the monkey, is $3121$. The link gives an outline of the solution to this classic Diophantine problem. A detailed solution can be found in this PDF, alone with the slick shortcut solution:

By inspection $-4$ is a solution: when it’s divided into $5$ piles of $-1$ with $1$ left over for the monkey, and one of the piles is removed, $-4$ coconuts remain for the next division. It’s also not too hard to see that $5^6=15625$ can be added to any solution to obtain another, since the pile is divided into $5$ piles $6$ times; thus, $15621$ is a solution.

Answer 2

Let $x_k$ be the number of coconuts left after the $k$-th sailor has divided them into $5$ piles, hidden his pile, and given the remaining coconut to the monkey. Hence,

$$x_{k+1} = \frac 45 (x_k - 1) = \frac 45 x_k - \frac 45$$

and, thus,

$$x_k = \left(\frac 45\right)^k x_0 - \sum_{\ell=1}^k \left(\frac 45\right)^{\ell} = \cdots = (x_0 + 4) \left(\frac 45\right)^k - 4$$

After the $5$ sailors' interventions, one coconut is given to the monkey and the remaining ones are divided evenly in $5$ piles. Thus, $x_5 - 1$ is not merely a nonnegative integer, but also a multiple of $5$, i.e., there exists $m \in \mathbb N$ such that $x_5 - 1 = 5 m$. Thus, the initial number of coconuts is given by

$$x_0 = - 4 + \frac{5^6}{4^5} (m+1)$$

The first natural $m$ that makes $x_0$ integral is $m = 4^5 - 1$. The first admissible $x_0$ is thus $$x_0 = 5^6 - 4 = \color{blue}{15621}$$

Computing $x_0, x_1, \dots, x_5$ in Haskell,

λ> take 6 $ iterate (\x->(div (4*(x-1)) 5)) 15621
[15621,12496,9996,7996,6396,5116]

Note that $x_5 = 5116 = 1 + 5 \cdot 1023$.

Answer 3

Derivation: how does a mathematician change a lightbulb? Level the house thereby reducing it to a previously solved problem.

Solve it without feed the monkey(s) first, just n men no monkeys. It's clear that $n^{n+1}$ can be divided by each of the $n$ man and again in the morning. So $n^{n+1}$ works here. In fact once you have a solution to the original problem, any solution, you can add $n^{n+1}$ to it and get another solution to the original problem.

Find a number $x$ you can subtract $c$ from, $c$ is the number of monkeys, then divide what remains by $n$ and then multiply by $n-1$ but the final answer is what you started with $x$.

$\dfrac{(n+1)(x-c)}{n} = x$, so $x$ is $-c(n-1)$ and $-c(n-1)$ is a solution, though an abstract solution, it is negative. Once you add any multiple of $n^{n+1}$ you get all the solutions, for there are an infinite numbers of solutions.

$$ M \cdot [n^{n+1}] - c(n-1) $$ To show that all solution s look like this, assume you have any two solutions and show that their difference is divisible by $n$, namely $n+1$ times.

Answer 4

Retired Professor Lynn Garner (former chairman of the BYU Mathematics Faculty), has found a general analytical solution to an extension of this problem, in m men.

His solution (which I have published by his permission from our private correspondence) may be found here (my article also includes a few solutions to the standard problem in 5 men): https://aaabit.com/Coconuts

In outline—

Let $n = n_0$ be the initial number of coconuts gathered.

$$n_1={m−1\over m}(n−1)$$

$$…$$

$$n_m=\left({m−1\over m}\right)^{mn}− \left\{\left({m−1\over m}\right)^{m−1}+…+\left({m−1\over m}\right)^2+{m−1\over m} \right\}$$

Rewriting this expression, summing over the geometric progression using $1+r+r^2+…+r^{m−1}={1-r^m\over 1−r}$, we get the number of coconuts left in the morning which is a multiple of m. Then let q be a positive integer, and set:

$$n_m= \left({m−1\over m} \right)^{m}n − (m−1)\left( {m^m−(m-1)^m\over m^m} \right)=qm$$

Solving for n, we get:

$$n=1−m+m^m\left(m−1+qm\over (m−1)^m\right)$$

In order for n to be an integer, we only need the quotient to be an integer. So let r be an integer and set:

$${m−1+qm\over (m−1)^m}=r$$

Isolating the qm term, and noting that $m$ and $m−1$ are relatively prime; we deduce that for some s:

$$(m−1)^{m−1}r−1=sm$$

$$(m−1)^{m−1}r=sm+1$$

That is, some multiple of $(m−1)^{m−1}$ is also one more than a multiple of m. That such integers r and s exist follows from the fact that $m$ and $m−1$ are relatively prime. For example,

m=3 r=1 s=1
m=4 r=3 s=20
m=5 r=1 s=51
m=6 r=5 s=2604
m=7 r=1 s=6665
m=8 r=7 s=720600

etc. We can easily prove that $r=1$ when m is odd, whereas $r=m−1$ when m is even; since $m−1$ is congruent to $-1$, so to an even power (m odd) is congruent to $1$, and to an odd power (m even) is still $-1$, so needs another factor of $m-1$ to square it. Therefore:

$$n=rm^m−m+1$$ $$r=\begin{cases}m\ odd:1\\m\ even:m−1\end{cases}$$

Examples of minimum possible solutions, where other solutions may be given by adding some multiple of $m^{m+1}$:

m=1    n=1
m=2    n=3
m=3    n=25
m=4    n=765
m=5    n=3121
m=6    n=233275
m=7    n=823537
m=8    n=117440505
m=9    n=387420481

Answer 5

This solution will satisfy any number of sailors & coconuts:

$n$ is the number of sailors

$c$ is the number of coconuts given to the monkey by each sailor.

Initial number of coconuts = $\left(n^{n+1}\right)-c(n-1)$

Bala Nair