The first sailor counts the nuts and find that they cannot be divided by 5. Therefore he gives the monkey a nut, which operation diminishes the heap of nuts so that the number now allows division by 5. And the sailor takes his part of the nuts and hides it and goes to sleep again.
 

The next sailor finds the same situation: The number of the nuts in the heap is not divisible by 5 and therefore the monkey gets a nut and thereby the problem is solved for him. So it continues. The same procedure through all night. And in the morning, when dawn colours the sky red, the division by 5 now works like a miracle. The structure is thereby described. And now we proceed with trial and error.

If we try with -4, the operation of giving the monkey a postive nut decreases the heap to -5 nuts. Which allows the sailor to take his part (one negative nut) and then the heap again

is -4.

You can immediately see that the next sailor and all of them has exactly the same heap to start with, and that in the morning - if you allow the monkey a 6th positive nut - each sailor will have arrived at -2 nuts. Salomonic wisdom: the monkey will have 6 positive nuts and the sailors together 10 negative nuts, i.e. 4 negative nuts in toto.

Now, this is not the correct solution. The monkey has got one nut too many and you cannot have negative nuts in reality. The next possible solution in our process of trial and error is -4 + 5⁵ = 3121. And that works.

For me this way of thinking is more easy to accept if in front of my inner mind I place the face of an oldfashioned watch. First we turn the pointer a bit to the left of 12 (say 4 minutes) and then we let the clock tick until it reaches the target point, which happens after 3125 minutes.

Other solutions than negative thinking is described in Gardners book. One of these is based on painting 4 of the nuts blue instead of regarding them as negative.

And interestingly Gardner's book has as the chapter preceding the one with the monkey and the nuts a chapter about the Golden Mean (i.e. the relation j = 1.61803398...), which is intimately connected with beauty and the pentagram. (And Venus, the platonic bodies etc etc).

The chapter about the Golden Mean in fact emphasizes all this by including a picture: The Last Supper by Salvador Dali:

The windows are evidently part of a dodecahedron, like a pentagram in three dimensions. The timespace is just before death and rebirth.