I hope you're good at Math!

Relative

My thinking was that if you 'lend' the pile four coconuts then the pile is always divisible by five. For example we begin with 'x' coconuts and add four; we will call this 'a'. This is divided by five. It divides evenly but of course the one doing the dividing has one more coconut than he would have had, so we take one back. ergo he has a-1 coconuts.
If we stipulate that he cannot take any of the four 'ghost coconuts, there are still four of those in the pile and it will again divide by five. And again and again.

So the answer is 5^6 minus the four ghost coconuts; 15625 - 4 = 15621. As easy as that.

So I exchanged the monkey for four ghost coconuts and it all becomes - IMHO - easier.

So for the general statement that you requested, the answer ought to be - I say 'ought to be' because this is 'off the cuff' and I have not yet checked it - ought to be:
The number of men to the power of the number of divisions minus 'the number of men minus one'. It may be that this would not always be the smallest possible answer; I will need to look into that.

Wow Iacomus, I'm impressed - you are surely a great thinker.
I knew you had the right idea with ghost coconuts - just wanted you to state it right. If we write the expression now:

N ^(N+1) - (N-1) is the smallest number!

The 'ghost' approach is great, and another explanation is as follows:
You imagine two piles, one of them only included in divisions, and another that is divided and gives one coconut each turn.
They are treated separately (since this is a linear problem), and clearly the large one must be divisable by number N N+1 - times. The smallest such number is N ^(N+1). Now the small pile is really your 'ghost pile' - and it contains a negative amount of coconuts - exactly -(N-1). First the monkey gets one coconut , thus decreasing small pile to -N. Then, the man takes 1/N, that is -1 coconut from it, and the small pile returns to the beginning state - and that is -(N-1).
In the end, ehough coconuts are left on the main pile not to get net negative result.

Looks like Iacomus and Dirac think alike

Relative

I would (humbly of course) question your general statement as being truly general.

Imagine that the five sailors were also suffering from acute amnesia. There is then nothing to stop each one getting up in the night and sharing out the coconuts again, having forgotten that he had already done so.

On the other hand one or more of the sailors might be either honest or a deep sleeper and do/does not divide the coconuts.

So if the number of sailors is 'n' then the number of divisions will not - necessarily - be 'n+1' and could be larger or smaller.


So, IMHO, we need two variables. Let the number of men be 'n' and let the number of divisions be 'd' then the 'truly general' statement would be
'n^d - (n-1)'
because there seems no reason that we can assume
'd =n+1' is true in every case.



If Dirac and Iacomus think alike - and I thank you for saying we do - I regret to say that it is only true as regards this one problem.

Alas that it is so, but it is.

P.S. 'ghost coconuts' and 'negative coconuts' gives coconuts an unexpectedly surreal aspect. This really appeals to my sense of humor

Iacomus - you're absolutely right in saying that

N^d - (N-1) is the solution to the more general problem of N men and d divisions!

There is, however, an even more general problem: suppose the monkey now takes not one, but three coconuts! The bastard monkey!

Now the solution is

N^d - m*(N-1)

where m is the number of coconuts that monkey gets each time! This will work as long as m is small enough. It is interesting to see that the more the monkey takes, the smaller the initial number of coconuts is!

Relative

You overlook the advantages of rank among these sailors. I suspect, if it is like any military organisation I ever saw, that senior officers would get x% more coconuts than junior officers who would get y% more then senior NCOs who would in turn get z% more than junior NCOs and the other ranks would have to make do with v% less than the junior NCOs.

And you were worried about the monkey??