# Sailors, coconuts and a monkey problem

**Sailors, coconuts and a monkey problem**

You are encouraged to solve this task according to the task description, using any language you may know.

Five sailors are shipwrecked on an island and collect a large pile of coconuts during the day.

That night the first sailor wakes up and decides to take his first share early so tries to divide the pile of coconuts equally into five piles but finds that there is one coconut left over, so he tosses it to a monkey and then hides "his" one of the five equally sized piles of coconuts and pushes the other four piles together to form a single visible pile of coconuts again and goes to bed.

To cut a long story short, each of the sailors in turn gets up once during the night and performs the same actions of dividing the coconut pile into five, finding that one coconut is left over and giving that single remainder coconut to the monkey.

In the morning (after the surreptitious and separate action of each of the five sailors during the night), the remaining coconuts are divided into five equal piles for each of the sailors, whereupon it is found that the pile of coconuts divides equally amongst the sailors with no remainder. (Nothing for the monkey in the morning.)

- The task

- Calculate the minimum possible size of the initial pile of coconuts collected during the first day.
- Use a method that assumes an answer is possible, and then applies the constraints of the tale to see if it is correct. (I.e. no applying some formula that generates the correct answer without integer divisions and remainders and tests on remainders; but constraint solvers
*are*allowed.) - Calculate the size of the initial pile of coconuts if six sailors were marooned and went through a similar process (but split into six piles instead of five of course).
- Show your answers here.

- Extra credit (optional)

- Give some indication of the number of coconuts each sailor hides during the night.

- Note

- Of course the tale is told in a world where the collection of any amount of coconuts in a day and multiple divisions of the pile, etc can occur in time fitting the story line, so as not to affect the mathematics.
- The tale is also told in a version where the monkey also gets a coconut in the morning. This is
*not*that tale!

- C.f

- Monkeys and Coconuts - Numberphile (Video) Analytical solution.
- A002021 Pile of coconuts problem The On-Line Encyclopedia of Integer Sequences. (Although some of its references may use the alternate form of the tale).

## Contents

## Bc[edit]

This script implements a solution in the coconuts function for a number of sailors > 1 and a number of monkeys between 1 and sailors-1. It also executes the coconuts function for some values of sailors/monkeys.

define coconuts(sailors, monkeys) {

print "coconuts(", sailors, ", ", monkeys, ") = "

if (sailors < 2 || monkeys < 1 || sailors <= monkeys) {

return 0

}

blue_cocos = sailors-1

pow_bc = blue_cocos^sailors

x_cocos = pow_bc

while ((x_cocos-blue_cocos)%sailors || (x_cocos-blue_cocos)/sailors < 1) {

x_cocos += pow_bc

}

return (x_cocos/pow_bc*(sailors^sailors)-blue_cocos)*monkeys

}

scale = 0

coconuts(1, 1)

coconuts(2, 1)

coconuts(3, 1)

coconuts(3, 2)

coconuts(4, 1)

coconuts(5, 1)

coconuts(5, 4)

coconuts(6, 1)

coconuts(101, 1)

- Output:

$ time bc <coconuts_bc.in coconuts(1, 1) = 0 coconuts(2, 1) = 11 coconuts(3, 1) = 25 coconuts(3, 2) = 50 coconuts(4, 1) = 765 coconuts(5, 1) = 3121 coconuts(5, 4) = 12484 coconuts(6, 1) = 233275 coconuts(101, 1) = 2731861967715741354199866657915606142014717766608\ 81280465910305960827252944980667223385057449021203688309007889238399\ 91099564447458450075226030128555294655577015766113909738825769262480\ 452415909200510001 real 0m0.141s user 0m0.031s sys 0m0.062s

## Befunge[edit]

This is a translation of the second C solution. The output lists the number of sailors, the size of the original pile, and the final share each sailor receives the following morning.

>2+:01p9>`#@_00v

nvg10*g10:+>#1$<

#>\:01g1-%#^_:0v

-|:-1\+1<+/-1g1<

1>$01g.">-",,48v

^g10,+55<.,9.,*<

- Output:

2 -> 11 1 3 -> 25 2 4 -> 765 60 5 -> 3121 204 6 -> 233275 13020 7 -> 823537 39990 8 -> 117440505 5044200 9 -> 387420481 14913080

## Bracmat[edit]

(Though without the`assert`

procedure.)
( ( divmod

= a b

. !arg:(?a.?b)&(div$(!a.!b).mod$(!a.!b))

)

& ( overnight

= ns nn result s q r

. !arg:(?ns.?nn)

& :?result

& 0:?s

& whl

' ( !s+1:?s:~>!ns

& divmod$(!nn.!ns):(?q.?r)

& !r:1

& !q*(!ns+-1):?nn

& !result (!s.!q.!r.!nn):?result

)

& !s:>!ns

& divmod$(!nn.!ns):(?q.0)

& !result

)

& ( minnuts

= nsailors nnuts result sailor takes gives leaves

. !arg:?nsailors

& 0:?nnuts

& whl

' ( 1+!nnuts:?nnuts

& ~(overnight$(!nsailors.!nnuts):?result)

)

& out$(!nsailors ": " !nnuts)

& whl

' ( !result:(?sailor.?takes.?gives.?leaves) ?result

& out

$ ( str

$ ( " Sailor #"

!sailor

" takes "

!takes

", giving "

!gives

" to the monkey and leaves "

!leaves

)

)

)

& out

$ ( str

$ ("In the morning, each sailor gets " !leaves*!nsailors^-1 " nuts")

)

)

& 4:?n

& whl

' ( 1+!n:~>6:?n

& out$("Solution with " !n " sailors:")

& minnuts$!n

)

)

Output:

Solution with 5 sailors: 5 : 3121 Sailor #1 takes 624, giving 1 to the monkey and leaves 2496 Sailor #2 takes 499, giving 1 to the monkey and leaves 1996 Sailor #3 takes 399, giving 1 to the monkey and leaves 1596 Sailor #4 takes 319, giving 1 to the monkey and leaves 1276 Sailor #5 takes 255, giving 1 to the monkey and leaves 1020 In the morning, each sailor gets 204 nuts Solution with 6 sailors: 6 : 233275 Sailor #1 takes 38879, giving 1 to the monkey and leaves 194395 Sailor #2 takes 32399, giving 1 to the monkey and leaves 161995 Sailor #3 takes 26999, giving 1 to the monkey and leaves 134995 Sailor #4 takes 22499, giving 1 to the monkey and leaves 112495 Sailor #5 takes 18749, giving 1 to the monkey and leaves 93745 Sailor #6 takes 15624, giving 1 to the monkey and leaves 78120 In the morning, each sailor gets 13020 nuts

## C[edit]

#include <stdio.h>

int valid(int n, int nuts)

{

int k;

for (k = n; k; k--, nuts -= 1 + nuts/n)

if (nuts%n != 1) return 0;

return nuts && !(nuts%n);

}

int main(void)

{

int n, x;

for (n = 2; n < 10; n++) {

for (x = 0; !valid(n, x); x++);

printf("%d: %d\n", n, x);

}

return 0;

}

- Output:

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

But it's faster to search backwards: if everyone receives some coconuts, see if we can backtrack to the original pile:

#include <stdio.h>

// calculates if everyone got some nuts in the end, what was the original pile

// returns 0 if impossible

int total(int n, int nuts)

{

int k;

for (k = 0, nuts *= n; k < n; k++) {

if (nuts % (n-1)) return 0;

nuts += nuts / (n-1) + 1;

}

return nuts;

}

int main(void)

{

int n, x, t;

for (n = 2; n < 10; n++) {

for (x = 1, t = 0; !(t = total(n, x)); x++);

printf("%d: %d\t%d\n", n, t, x);

}

return 0;

}

- Output:

sailers: original pile, final share

2: 11 1 3: 25 2 4: 765 60 5: 3121 204 6: 233275 13020 7: 823537 39990 8: 117440505 5044200 9: 387420481 14913080

## C#[edit]

class Test

{

static bool valid(int n, int nuts)

{

for (int k = n; k != 0; k--, nuts -= 1 + nuts / n)

{

if (nuts % n != 1)

{

return false;

}

}

return nuts != 0 && (nuts % n == 0);

}

static void Main(string[] args)

{

int x = 0;

for (int n = 2; n < 10; n++)

{

while (!valid(n, x))

x++;

System.Console.WriteLine(n + ": " + x);

}

}

}

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

## Elixir[edit]

### Brute Force[edit]

defmodule RC do

def valid?(sailor, nuts), do: valid?(sailor, nuts, sailor)

def valid?(sailor, nuts, 0), do: nuts > 0 and rem(nuts,sailor) == 0

def valid?(sailor, nuts, _) when rem(nuts,sailor)!=1, do: false

def valid?(sailor, nuts, i) do

valid?(sailor, nuts - div(nuts,sailor) - 1, i-1)

end

end

Enum.each([5,6], fn sailor ->

nuts = Enum.find(Stream.iterate(sailor, &(&1+1)), fn n -> RC.valid?(sailor, n) end)

IO.puts "\n#{sailor} sailors => #{nuts} coconuts"

Enum.reduce(0..sailor, nuts, fn _,n ->

{d, r} = {div(n,sailor), rem(n,sailor)}

IO.puts " #{inspect [n, d, r]}"

n - 1 - d

end)

end)

- Output:

5 sailors => 3121 coconuts [3121, 624, 1] [2496, 499, 1] [1996, 399, 1] [1596, 319, 1] [1276, 255, 1] [1020, 204, 0] 6 sailors => 233275 coconuts [233275, 38879, 1] [194395, 32399, 1] [161995, 26999, 1] [134995, 22499, 1] [112495, 18749, 1] [93745, 15624, 1] [78120, 13020, 0]

### Faster version[edit]

defmodule Sailor do

def coconuts(sailor), do: coconuts(sailor, sailor)

defp coconuts(sailor, nuts) do

if n = do_coconuts(sailor, nuts, sailor), do: n, else: coconuts(sailor, nuts+sailor)

end

defp do_coconuts(_sailor, nuts, 0), do: nuts

defp do_coconuts(sailor, nuts, _) when rem(nuts, sailor-1) != 0, do: nil

defp do_coconuts(sailor, nuts, i) do

do_coconuts(sailor, nuts + div(nuts, sailor-1) + 1, i-1)

end

end

Enum.each(2..9, fn sailor ->

IO.puts "#{sailor}: #{Sailor.coconuts(sailor)}"

end)

- Output:

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

## Forth[edit]

: total

over * over 1- rot 0 ?do

over over mod if dup xor swap leave else over over / 1+ rot + swap then

loop drop

;

: sailors

1+ 2 ?do

1 begin i over total dup 0= while drop 1+ repeat cr i 0 .r ." : " . .

loop

;

9 sailors

- Output:

2: 11 1 3: 25 2 4: 765 60 5: 3121 204 6: 233275 13020 7: 823537 39990 8: 117440505 5044200 9: 387420481 14913080 ok

## Haskell[edit]

This program works by applying a function to increasing multiples of the number of sailors. The function takes a potential final number of coconuts (at the time the sailors awaken) and works backwards to get to the initial number of coconuts. At every step, it will abort the computation if the current number of coconuts can't arise as a result of splitting the previous pile.

import Control.Monad

import Data.Maybe

import System.Environment

-- Takes the number of sailors and the final number of coconuts. Returns

-- Just the associated initial number of coconuts and Nothing otherwise.

tryFor :: Int -> Int -> Maybe Int

tryFor s = foldr (>=>) pure $ replicate s step

where step n

| n `mod` (s - 1) == 0 = Just $ n * s `div` (s - 1) + 1

| otherwise = Nothing

-- Gets the number of sailors from the first command-line argument and

-- assumes 5 as a default if none is given. Then uses tryFor to find the

-- smallest solution.

main :: IO ()

main = do

args <- getArgs

let n = case args of

[] -> 5

s:_ -> read s

a = head . mapMaybe (tryFor n) $ [n, 2 * n ..]

print a

Examples:

$ ./coconuts 3121 $ ./coconuts 4 765 $ ./coconuts 6 233275

## J[edit]

Here, we assume an answer which is less than 10000, and try each possibility, constraining ourselves to the list of answers which are valid. (As it happens, there's only one of those.)

I.(=<.)%&5 verb def'4*(y-1)%5'^:5 i.10000

3121

These sailors must count coconuts extremely quickly.

When we do this with six sailors, it turns out that we have to assume a larger initial value:

I.(=<.)%&6 verb def'5*(y-1)%6'^:6 i.1000000

233275 513211 793147

If it were not obvious which of the answers here was the minimum value we could additionally pick the smallest value from the list. But that would require typing two extra characters (for example, replace I. with i.&1), and most people already have so much trouble reading J that the extra code would surely be too much.

## Java[edit]

public class Test {

static boolean valid(int n, int nuts) {

for (int k = n; k != 0; k--, nuts -= 1 + nuts / n)

if (nuts % n != 1)

return false;

return nuts != 0 && (nuts % n == 0);

}

public static void main(String[] args) {

int x = 0;

for (int n = 2; n < 10; n++) {

while (!valid(n, x))

x++;

System.out.printf("%d: %d%n", n, x);

}

}

}

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

## JavaScript[edit]

### ES5[edit]

( As in the recursive Python example )

(function () {

// wakeSplit :: Int -> Int -> Int -> Int

function wakeSplit(intNuts, intSailors, intDepth) {

var nDepth = intDepth !== undefined ? intDepth : intSailors,

portion = Math.floor(intNuts / intSailors),

remain = intNuts % intSailors;

return 0 >= portion || remain !== (nDepth ? 1 : 0) ?

null : nDepth ? wakeSplit(

intNuts - portion - remain, intSailors, nDepth - 1

) : intNuts;

}

// TEST for 5, 6, and 7 intSailors

return [5, 6, 7].map(function (intSailors) {

var intNuts = intSailors;

while (!wakeSplit(intNuts, intSailors)) intNuts += 1;

return intNuts;

});

})();

- Output:

[3121, 233275, 823537]

### ES6[edit]

Adding just a touch of curry to the coconuts. (See the Rosetta Code task: Currying)

(() => {

// wakeSplit :: Int -> Int -> Int -> Int

let wakeSplit = (intSailors, intNuts, intDepth) => {

let nDepth = intDepth !== undefined ? intDepth : intSailors,

portion = Math.floor(intNuts / intSailors),

remain = intNuts % intSailors;

return 0 >= portion || remain !== (nDepth ? 1 : 0) ?

null : nDepth ? wakeSplit(

intSailors, intNuts - portion - remain, nDepth - 1

) : intNuts;

};

//GENERIC FUNCTIONS

// curry :: ((a, b) -> c) -> a -> b -> c

let curry = f => a => b => f(a, b),

// until :: (a -> Bool) -> (a -> a) -> a -> a

until = (p, f, x) => {

let v = x;

while (!p(v)) v = f(v);

return v;

},

// succ :: Int -> Int

succ = x => x + 1;

// TEST for 5, 6, and 7 Sailors

return [5, 6, 7].map(intSailors => {

let intNuts = intSailors,

test = curry(wakeSplit)(intSailors);

return until(test, succ, intNuts);

});

})();

- Output:

[3121, 233275, 823537]

## jq[edit]

The first solution presented in this section is based on a simulation of the nighttime and daytime activities, and the second (much faster) solution works backwards.

If your jq does not have "until" defined, here is its definition:

def until(cond; next): def _until: if cond then . else (next|_until) end; _until;

### Simulation[edit]

# If n (the input) is an admissible number of coconuts with respect to

# the night-time squirreling away of the coconuts by "sailors" sailors, then give 1 to the

# monkey, let one sailor squirrel away (1/sailors) coconuts, and yield the remaining number;

# otherwise, return false:

def squirrel(sailors):

def admissible: if . then (. % sailors) == 1 else . end;

if admissible then . - ((. - 1) / sailors) - 1

else false

end;

def nighttime(sailors):

reduce range(0; sailors) as $i (.; squirrel(sailors));

def morning(sailors):

if . then (. % sailors) == 0

else false

end;

# Test whether the input is a valid number of coconuts with respect to the story:

def valid(sailors): nighttime(sailors) | morning(sailors);

**Five sailors**:
Find the minimum number of coconuts if there are 5 sailors --
start at 1 as there must be at least one to give to the monkey during the night:

1 | until( valid(5); . + 1)

- Output:

3121

**Six sailors**:
Find the minimum number of coconuts if there are 6 sailors --
start at 1 as there must be at least one to give to the monkey during the night:

1 | until( valid(6); . + 1)

- Output:

233275

### Working backwards[edit]

# If n (the input) is the number of coconuts remaining after

# the surreptitious squirreling away by one sailor,

# then emit the number of coconuts which that sailor originally

# saw if n is admissible, otherwise emit false:

def unsquirrel(sailors):

if . and (. % (sailors - 1) == 0)

then 1 + (sailors * (. / (sailors - 1)))

else false

end;

# If in the end each sailor received n coconuts (where n is the input), how many coconuts

# were there initially?

def backwards(sailors):

reduce range(0; sailors) as $i (. * sailors; unsquirrel(sailors));

def solve:

. as $sailors

# state: [ final_number_per_sailor, original_number_of_coconuts]

| [-1] | until( .[1]; .[0] += 1 | .[1] = (.[0] | backwards($sailors)) )

| "With \($sailors) sailors, there were originally \(.[1]) coconuts,"+

" and each sailor finally ended up with \(.[0])." ;

range(2;9) | solve

- Output:

With 2 sailors, there were originally 3 coconuts, and each sailor finally ended up with 0.

With 3 sailors, there were originally 25 coconuts, and each sailor finally ended up with 2.

With 4 sailors, there were originally 765 coconuts, and each sailor finally ended up with 60.

With 5 sailors, there were originally 3121 coconuts, and each sailor finally ended up with 204.

With 6 sailors, there were originally 233275 coconuts, and each sailor finally ended up with 13020.

With 7 sailors, there were originally 823537 coconuts, and each sailor finally ended up with 39990.

With 8 sailors, there were originally 117440505 coconuts, and each sailor finally ended up with 5044200.

## Perl 6[edit]

There is nowhere in the spec where it explicitly states that the sailors cannot equally share zero coconuts in the morning. Actually, The On-Line Encyclopedia of Integer Sequences A002021 considers the cases for 1 and 2 sailors equally sharing zero coconuts in the morning to be the correct answer.

This will test combinations of sailors and coconuts to see if they form a valid pairing. The first 6 are done using brute force, testing every combination until a valid one is found. For cases of 7 to 15 sailors, it uses a carefully crafted filter to drastically reduce the number of trials needed to find a valid case (to one, as it happens... :-) )

my @ones = flat 'th', 'st', 'nd', 'rd', 'th' xx 6;

my @teens = 'th' xx 10;

my @suffix = lazy flat (@ones, @teens, @ones xx 8) xx *;

# brute force the first six

for 1 .. 6 -> $sailors { for $sailors .. * -> $coconuts { last if check( $sailors, $coconuts ) } }

# finesse 7 through 15

for 7 .. 15 -> $sailors { next if check( $sailors, coconuts( $sailors ) ) }

sub is_valid ( $sailors is copy, $nuts is copy ) {

return 0, 0 if $sailors == $nuts == 1;

my @shares;

for ^$sailors {

return () unless $nuts % $sailors == 1;

push @shares, ($nuts - 1) div $sailors;

$nuts -= (1 + $nuts div $sailors);

}

push @shares, $nuts div $sailors;

return @shares if !?($nuts % $sailors);

}

sub check ($sailors, $coconuts) {

if my @piles = is_valid($sailors, $coconuts) {

say "\nSailors $sailors: Coconuts $coconuts:";

for ^(@piles - 1) -> $k {

say "[email protected][$k+1] takes @piles[$k], gives 1 to the monkey."

}

say "The next morning, each sailor takes @piles[*-1]\nwith none left over for the monkey.";

return True;

}

False;

}

multi sub coconuts ( $sailors where { $sailors % 2 == 0 } ) { ($sailors - 1) * ($sailors ** $sailors - 1) }

multi sub coconuts ( $sailors where { $sailors % 2 == 1 } ) { $sailors ** $sailors - $sailors + 1 }

- Output:

Sailors 1: Coconuts 1: 1st takes 0, gives 1 to the monkey. The next morning, each sailor takes 0 with none left over for the monkey. Sailors 2: Coconuts 3: ... Sailors 5: Coconuts 3121: 1st takes 624, gives 1 to the monkey. 2nd takes 499, gives 1 to the monkey. 3rd takes 399, gives 1 to the monkey. 4th takes 319, gives 1 to the monkey. 5th takes 255, gives 1 to the monkey. The next morning, each sailor takes 204 with none left over for the monkey. Sailors 6: Coconuts 233275: 1st takes 38879, gives 1 to the monkey. 2nd takes 32399, gives 1 to the monkey. 3rd takes 26999, gives 1 to the monkey. 4th takes 22499, gives 1 to the monkey. 5th takes 18749, gives 1 to the monkey. 6th takes 15624, gives 1 to the monkey. The next morning, each sailor takes 13020 with none left over for the monkey. Sailors 7: Coconuts 823537: ... Sailors 15: Coconuts 437893890380859361: ...

## Phix[edit]

The morning pile must be a multiple of sailors, so this only tries multiples of sailors! Needed an ugly kludge for solve(1), the limit of 1 billion suffices for solve(9), above that gets run-time type check errors as capacity of ints are blown anyway.

procedure solve(integer sailors)

integer m, sm1 = sailors-1

if sm1=0 then -- edge condition for solve(1) [ avoid /0 ]

m = sailors

else

for n=sailors to 1_000_000_000 by sailors do -- morning pile divisible by #sailors

m = n

for j=1 to sailors do -- see if all of the sailors could..

if remainder(m,sm1)!=0 then -- ..have pushed together sm1 piles

m = 0 -- (no: try a higher n)

exit

end if

m = sailors*m/sm1+1 -- add sailor j's stash and one for the monkey

end for

if m!=0 then exit end if

end for

end if

printf(1,"Solution with %d sailors: %d\n",{sailors,m})

for i=1 to sailors do

m -= 1 -- one for the monkey

m /= sailors

printf(1,"Sailor #%d takes %d, giving 1 to the monkey and leaving %d\n",{i,m,m*sm1})

m *= (sm1)

end for

printf(1,"In the morning each sailor gets %d nuts\n",m/sailors)

end procedure

solve(5)

solve(6)

- Output:

Solution with 5 sailors: 3121 Sailor #1 takes 624, giving 1 to the monkey and leaving 2496 Sailor #2 takes 499, giving 1 to the monkey and leaving 1996 Sailor #3 takes 399, giving 1 to the monkey and leaving 1596 Sailor #4 takes 319, giving 1 to the monkey and leaving 1276 Sailor #5 takes 255, giving 1 to the monkey and leaving 1020 In the morning each sailor gets 204 nuts Solution with 6 sailors: 233275 Sailor #1 takes 38879, giving 1 to the monkey and leaving 194395 Sailor #2 takes 32399, giving 1 to the monkey and leaving 161995 Sailor #3 takes 26999, giving 1 to the monkey and leaving 134995 Sailor #4 takes 22499, giving 1 to the monkey and leaving 112495 Sailor #5 takes 18749, giving 1 to the monkey and leaving 93745 Sailor #6 takes 15624, giving 1 to the monkey and leaving 78120 In the morning each sailor gets 13020 nuts

## Python[edit]

You may want to read Solving the Monkey and coconuts problem to get more background on the evolution of the Python code.

### Python: Procedural[edit]

def monkey_coconuts(sailors=5):

"Parameterised the number of sailors using an inner loop including the last mornings case"

nuts = sailors

while True:

n0, wakes = nuts, []

for sailor in range(sailors + 1):

portion, remainder = divmod(n0, sailors)

wakes.append((n0, portion, remainder))

if portion <= 0 or remainder != (1 if sailor != sailors else 0):

nuts += 1

break

n0 = n0 - portion - remainder

else:

break

return nuts, wakes

if __name__ == "__main__":

for sailors in [5, 6]:

nuts, wake_stats = monkey_coconuts(sailors)

print("\nFor %i sailors the initial nut count is %i" % (sailors, nuts))

print("On each waking, the nut count, portion taken, and monkeys share are:\n ",

',\n '.join(repr(ws) for ws in wake_stats))

- Output:

For 5 sailors the initial nut count is 3121 On each waking, the nut count, portion taken, and monkeys share are: (3121, 624, 1), (2496, 499, 1), (1996, 399, 1), (1596, 319, 1), (1276, 255, 1), (1020, 204, 0) For 6 sailors the initial nut count is 233275 On each waking, the nut count, portion taken, and monkeys share are: (233275, 38879, 1), (194395, 32399, 1), (161995, 26999, 1), (134995, 22499, 1), (112495, 18749, 1), (93745, 15624, 1), (78120, 13020, 0)

### Python: Recursive[edit]

def wake_and_split(n0, sailors, depth=None):

if depth is None:

depth = sailors

portion, remainder = divmod(n0, sailors)

if portion <= 0 or remainder != (1 if depth else 0):

return None

else:

return n0 if not depth else wake_and_split(n0 - portion - remainder, sailors, depth - 1)

def monkey_coconuts(sailors=5):

"Parameterised the number of sailors using recursion including the last mornings case"

nuts = sailors

while True:

if wake_and_split(n0=nuts, sailors=sailors) is None:

nuts += 1

else:

break

return nuts

if __name__ == "__main__":

for sailors in [5, 6]:

nuts = monkey_coconuts(sailors)

print("For %i sailors the initial nut count is %i" % (sailors, nuts))

- Output:

For 5 sailors the initial nut count is 3121 For 6 sailors the initial nut count is 233275

### by solving Diophantine equation[edit]

The following is a more or less general solution for arbitrary number of sailors and varying numbers of coconuts the monkey gets. The monkey can be given more coconuts than there are sailors each turn. This is not part of task requirement.

# gives one solution of (x,y) for a x + by = c

def dioph(a, b, c):

aa,bb,x,y = a, b, 0, 1

while True:

q,a,b = a//b, b, a%b

x,y = y - q*x, x

if abs(a) == 1: break

if y*aa % bb != 1: y = -y

x,y = y*c, (c - aa*y*c)//bb

#assert(x*aa + y*bb == c)

return x,y

# rems: what monkey got each turn

# min_share: each sailor needs to get at least this many in the final round

def calcnuts(rems, min_share = 0):

n, r = len(rems) - 1, 0

c = (n - 1)**n

for x in rems: r,c = r + x*c, c//(n-1)*n

a, b = (n-1)**n, n**(n+1)

x, y = dioph(a, -b, r)

k = (min_share - y + a - 1)//a

return x + k*b, y + k*a

def distribute(nuts, monkey_nuts):

n = len(monkey_nuts) - 1

print("\n%d sailors, %d nuts:"%(n, nuts))

for r in monkey_nuts[:-1]:

p = (nuts - r)//n

print("\tNuts %d, hide %d, monkey gets %d" % (nuts, p, r))

nuts = p*(n - 1)

r = monkey_nuts[-1]

p = (nuts - r)//n

print("Finally:\n\tNuts %d, each share %d, monkey gets %d" % (nuts, p, r))

for sailors in range(2, 10):

monkey_loot = [1]*sailors + [0]

distribute(calcnuts(monkey_loot, 1)[0], monkey_loot)

# many sailors, many nuts

#for i in range(1, 5): print(10**i, calcnuts([1]*10**i + [0])[0])

## Racket[edit]

#lang racket

(define (wake-and-split nuts sailors depth wakes)

(define-values (portion remainder) (quotient/remainder nuts sailors))

(define monkey (if (zero? depth) 0 1))

(define new-wakes (cons (list nuts portion remainder) wakes))

(and (positive? portion)

(= remainder monkey)

(if (zero? depth)

new-wakes

(wake-and-split (- nuts portion remainder) sailors (sub1 depth) new-wakes))))

(define (sleep-and-split nuts sailors)

(wake-and-split nuts sailors sailors '()))

(define (monkey_coconuts (sailors 5))

(let loop ([nuts sailors])

(or (sleep-and-split nuts sailors)

(loop (add1 nuts)))))

(for ([sailors (in-range 5 7)])

(define wakes (monkey_coconuts sailors))

(printf "For ~a sailors the initial nut count is ~a\n" sailors (first (last wakes)))

(map displayln (reverse wakes))

(newline))

- Output:

For 5 sailors the initial nut count is 3121 (3121 624 1) (2496 499 1) (1996 399 1) (1596 319 1) (1276 255 1) (1020 204 0) For 6 sailors the initial nut count is 233275 (233275 38879 1) (194395 32399 1) (161995 26999 1) (134995 22499 1) (112495 18749 1) (93745 15624 1) (78120 13020 0)

## REXX[edit]

### uses a subroutine[edit]

{from the 1^{st}

**C**example}

/*REXX program solves a riddle of 5 sailors, a pile of coconuts, and a monkey. */

parse arg L H .; if L=='' then L=5 /*L not specified? Then use default.*/

if H=='' then H=6 /*H " " " " default.*/

/*{Tars is an old name for sailors.} */

do n=L to H /*traipse through a number of sailors. */

do $=0 while \valid(n,$); end /*perform while not valid coconuts. */

say 'sailors='n " coconuts="$ /*display number of sailors & coconuts.*/

end /*n*/

exit /*stick a fork in it, we're all done. */

/*──────────────────────────────────────────────────────────────────────────────────────*/

valid: procedure; parse arg n,nuts /*obtain the number sailors & coconuts.*/

do k=n by -1 for n /*step through the possibilities. */

if nuts//n\==1 then return 0 /*Not one coconut left? No solution. */

nuts=nuts - (1+nuts%n) /*subtract number of coconuts from pile*/

end /*k*/

return (nuts\==0) & \(nuts//n\==0) /*see if number coconuts>0 & remainder.*/

Programming note: The parentheses in the last REXX (**return**) statement aren't necessary, but help for readability.

**output** when using the default inputs:

sailors=5 coconuts=3121 sailors=6 coconuts=233275

### uses in-line code[edit]

This REXX version is the same as the above version (but the defaults are different),

and it also eliminates the use of a subroutine, making it faster.

/*REXX program solves a riddle of 5 sailors, a pile of coconuts, and a monkey. */

do n=2 to 9 /*traipse through number of sailors. */

do $=0; nuts=$ /*perform while not valid # coconuts. */

do k=n by -1 for n /*step through the possibilities. */

if nuts//n\==1 then iterate $ /*Not one coconut left? No solution.*/

nuts=nuts - (1+nuts%n) /*subtract number of coconuts from pile*/

end /*k*/

if (nuts\==0) & \(nuts//n\==0) then leave /*is this a solution to the riddle ? */

end /*$*/

say 'sailors='n " coconuts="$ /*display number of sailors & coconuts.*/

end /*n*/ /*stick a fork in it, we're all done. */

**output** when using the default inputs:

sailors=2 coconuts=11 sailors=3 coconuts=25 sailors=4 coconuts=765 sailors=5 coconuts=3121 sailors=6 coconuts=233275 sailors=7 coconuts=823537 sailors=8 coconuts=117440505 sailors=9 coconuts=387420481

### [edit]

{from the 2^{nd}

**C**example}

/*REXX program solves a riddle of 5 sailors, a pile of coconuts, and a monkey. */

parse arg L H .; if L=='' then L=2 /*L not specified? Then use default.*/

if H=='' then H=9 /*H " " " " " */

/*{Tars is an old name for sailors.} */

do n=L to H /*traipse through the number of sailors*/

do $=1 until t\==0 /*perform while number coconuts not 0. */

t=total(n,$) /*perform while not valid coconuts. */

end /*$*/

say 'sailors='n " coconuts="t ' share='$

end /*n*/

exit /*stick a fork in it, we're all done. */

/*──────────────────────────────────────────────────────────────────────────────────────*/

total: procedure; parse arg n,nuts /*obtain the number sailors & coconuts.*/

nuts=nuts*n

nn=n-1 /*NN is used as calculation shortcut. */

do k=0 for n /*step through the possibilities. */

if nuts//nn\==0 then return 0 /*Not one coconut left? No solution. */

nuts=nuts + nuts%nn + 1 /*bump the number coconuts to the pile.*/

end /*k*/

return nuts /*see if number coconuts>0 & remainder.*/

**output** when using the default inputs:

sailors=2 coconuts=11 share=1 sailors=3 coconuts=25 share=2 sailors=4 coconuts=765 share=60 sailors=5 coconuts=3121 share=204 sailors=6 coconuts=233275 share=13020 sailors=7 coconuts=823537 share=39990 sailors=8 coconuts=117440505 share=5044200 sailors=9 coconuts=387420481 share=14913080

## Ruby[edit]

### Brute Force[edit]

def valid?(sailor, nuts)

sailor.times do

return false if (nuts % sailor) != 1

nuts -= 1 + nuts / sailor

end

nuts > 0 and nuts % sailor == 0

end

[5,6].each do |sailor|

n = sailor

n += 1 until valid?(sailor, n)

puts "\n#{sailor} sailors => #{n} coconuts"

(sailor+1).times do

div, mod = n.divmod(sailor)

puts " #{[n, div, mod]}"

n -= 1 + div

end

end

- Output:

5 sailors => 3121 coconuts [3121, 624, 1] [2496, 499, 1] [1996, 399, 1] [1596, 319, 1] [1276, 255, 1] [1020, 204, 0] 6 sailors => 233275 coconuts [233275, 38879, 1] [194395, 32399, 1] [161995, 26999, 1] [134995, 22499, 1] [112495, 18749, 1] [93745, 15624, 1] [78120, 13020, 0]

### Faster version[edit]

def coconuts(sailor)

sailor.step(by:sailor) do |nuts|

flag = sailor.times do

break if nuts % (sailor-1) != 0

nuts += nuts / (sailor-1) + 1

end

return nuts if flag

end

end

(2..9).each do |sailor|

puts "#{sailor}: #{coconuts(sailor)}"

end

- Output:

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

### Implementation of Analysis on Discussion Page[edit]

def ng (sailors)

def _ng (sailors, iter, start)

n, g = [start], [start/sailors]

(1..iter).each{|s|

g[s],rem = n[s-1].divmod(sailors-1)

rem > 0 ? (return false) : n[s] = g[s]*sailors + 1

}

return [n,g]

end

n, start, step = [], sailors*(sailors-1), 1

(2..sailors).each{|s|

g=0; until n=_ng(sailors,s,start + g*step*sailors*(sailors-1)) do g=g+1 end

start,step = n[0][0], step*(sailors-1)

}

return n

end

- Output:

(3..10).each{|sailors| puts "Number of sailors = #{sailors}"; p ng(sailors)}

Number of sailors = 3 [[6, 10, 16, 25], [2, 3, 5, 8]] Number of sailors = 4 [[240, 321, 429, 573, 765], [60, 80, 107, 143, 191]] Number of sailors = 5 [[1020, 1276, 1596, 1996, 2496, 3121], [204, 255, 319, 399, 499, 624]] Number of sailors = 6 [[78120, 93745, 112495, 134995, 161995, 194395, 233275], [13020, 15624, 18749, 22499, 26999, 32399, 38879]] Number of sailors = 7 [[279930, 326586, 381018, 444522, 518610, 605046, 705888, 823537], [39990, 46655, 54431, 63503, 74087, 86435, 100841, 117648]] Number of sailors = 8 [[40353600, 46118401, 52706745, 60236281, 68841465, 78675961, 89915385, 102760441, 117440505], [5044200, 5764800, 6588343, 7529535, 8605183, 9834495, 11239423, 12845055, 14680063]] Number of sailors = 9 [[134217720, 150994936, 169869304, 191102968, 214990840, 241864696, 272097784, 306110008, 344373760, 387420481], [14913080, 16777215, 18874367, 21233663, 23887871, 26873855, 30233087, 34012223, 38263751, 43046720]] Number of sailors = 10 [[31381059600, 34867844001, 38742048891, 43046720991, 47829689991, 53144099991, 59048999991, 65609999991, 72899999991, 80999999991, 89999999991], [3138105960, 3486784400, 3874204889, 4304672099, 4782968999, 5314409999, 5904899999, 6560999999, 7289999999, 8099999999, 8999999999]]

Did someone ask the value for 100 sailors?

n = ng(100)

(0..100).each{|g| puts "#{n[0][100-g]}:#{n[1][100-g]}"}

The number of coconuts requires is as follows, the whole output is at Sailors, coconuts and a monkey problem/Ruby output 100

9899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999901

## Sidef[edit]

func coconuts(sailors, monkeys=1) {

if ((sailors < 2) || (monkeys < 1) || (sailors <= monkeys)) {

return 0

}

var blue_cocos = sailors-1

var pow_bc = blue_cocos**sailors

var x_cocos = pow_bc

while ((x_cocos-blue_cocos)%sailors || ((x_cocos-blue_cocos)/sailors < 1)) {

x_cocos += pow_bc

}

return monkeys*(x_cocos / pow_bc * sailors**sailors - blue_cocos)

}

2.to(9).each { |sailor|

say "#{sailor}: #{coconuts(sailor)}";

}

- Output:

2: 11 3: 25 4: 765 5: 3121 6: 233275 7: 823537 8: 117440505 9: 387420481

## Tcl[edit]

This is a very straightforward implementation. The "overnight" proc attempts to fulfill the activities of the night, throwing an error (through "assert") if it cannot. "minnuts" keeps trying to call it with more nuts until it succeeds. On success, "overnight" will return a list which narrates the night's activities.

proc assert {expr {msg ""}} { ;# for "static" assertions that throw nice errors

if {![uplevel 1 [list expr $expr]]} {

if {$msg eq ""} {

catch {set msg "{[uplevel 1 [list subst -noc $expr]]}"}

}

throw {ASSERT ERROR} "{$expr} $msg"

}

}

proc divmod {a b} {

list [expr {$a / $b}] [expr {$a % $b}]

}

proc overnight {ns nn} {

set result {}

for {set s 0} {$s < $ns} {incr s} {

lassign [divmod $nn $ns] q r

assert {$r eq 1} "Incorrect remainder in round $s (expected 1, got $r)"

set nn [expr {$q*($ns-1)}]

lappend result $s $q $r $nn

}

lassign [divmod $nn $ns] q r

assert {$r eq 0} "Incorrect remainder at end (expected 0, got $r)"

return $result

}

proc minnuts {nsailors} {

while 1 {

incr nnuts

try {

set result [overnight $nsailors $nnuts]

} on error {} {

# continue

} on ok {} {

break

}

}

puts "$nsailors: $nnuts"

foreach {sailor takes gives leaves} $result {

puts " Sailor #$sailor takes $takes, giving $gives to the monkey and leaves $leaves"

}

puts "In the morning, each sailor gets [expr {$leaves/$nsailors}] nuts"

}

foreach n {5 6} {

puts "Solution with $n sailors:"

minnuts $n

}

- Output:

Solution with 5 sailors: 5: 3121 Sailor #0 takes 624, giving 1 to the monkey and leaves 2496 Sailor #1 takes 499, giving 1 to the monkey and leaves 1996 Sailor #2 takes 399, giving 1 to the monkey and leaves 1596 Sailor #3 takes 319, giving 1 to the monkey and leaves 1276 Sailor #4 takes 255, giving 1 to the monkey and leaves 1020 In the morning, each sailor gets 204 nuts Solution with 6 sailors: 6: 233275 Sailor #0 takes 38879, giving 1 to the monkey and leaves 194395 Sailor #1 takes 32399, giving 1 to the monkey and leaves 161995 Sailor #2 takes 26999, giving 1 to the monkey and leaves 134995 Sailor #3 takes 22499, giving 1 to the monkey and leaves 112495 Sailor #4 takes 18749, giving 1 to the monkey and leaves 93745 Sailor #5 takes 15624, giving 1 to the monkey and leaves 78120 In the morning, each sailor gets 13020 nuts

## uBasic/4tH[edit]

For performance reasons, we limit ourselves to seven sailors.

For n = 2 To 7

t = 0

For x = 1 Step 1 While t = 0

t = FUNC(_Total(n,x))

Next

Print n;": ";t;Tab(12); x - 1

Next

End

_Total Param(2)

Local(1)

b@ = b@ * a@

a@ = a@ - 1

For c@ = 0 To a@

If b@ % a@ Then

b@ = 0

Break

EndIf

b@ = b@ + 1 + b@ / a@

Next

Return (b@)

- Output:

2: 11 1 3: 25 2 4: 765 60 5: 3121 204 6: 233275 13020 7: 823537 39990 0 OK, 0:127

## zkl[edit]

fcn monkey_coconuts(sailors=5){

nuts,wakes:=sailors,List();

while(True){

n0:=nuts; wakes.clear();

foreach sailor in (sailors + 1){

portion, remainder := n0.divr(sailors);

wakes.append(T(n0, portion, remainder));

if(portion <= 0 or remainder != (sailor != sailors).toInt()){

nuts += 1;

break;

}

n0 = n0 - portion - remainder;

}

fallthrough{ break }

}

return(nuts, wakes)

}

foreach sailors in ([5..6]){

nuts, wake_stats := monkey_coconuts(sailors);

println("For %d sailors the initial nut count is %,d".fmt(sailors, nuts));

println("On each waking, the nut count, portion taken, and monkeys share are:\n ",

wake_stats.concat("\n "));

}

- Output:

For 5 sailors the initial nut count is 3,121 On each waking, the nut count, portion taken, and monkeys share are: L(3121,624,1) L(2496,499,1) L(1996,399,1) L(1596,319,1) L(1276,255,1) L(1020,204,0) For 6 sailors the initial nut count is 233,275 On each waking, the nut count, portion taken, and monkeys share are: L(233275,38879,1) L(194395,32399,1) L(161995,26999,1) L(134995,22499,1) L(112495,18749,1) L(93745,15624,1) L(78120,13020,0)

fcn total(n, nuts){

nuts *= n;

foreach k in (n){

if (nuts % (n-1)) return(0);

nuts += nuts / (n-1) + 1;

}

nuts;

}

println("sailers: original pile, final share");

foreach n,x in ([2..9],[1..]){

if(t := total(n, x)){

print("%d: %d\t%d\n".fmt(n, t, x));

break;

}

}

- Output:

sailers: original pile, final share 2: 11 1 3: 25 2 4: 765 60 5: 3121 204 6: 233275 13020 7: 823537 39990 8: 117440505 5044200 9: 387420481 14913080