## The monkey and negative number of coconuts

There’s this math puzzle that I recently heard from my friend. Here it goes:

### The puzzle

In the not so distant past, there was this small island. And there’s this monkey… what? Yes, it’s alone. Ok, fine, he’s alone. Yes, poor thing.

So, on this island, there were many many many coconut trees. And each of them bore enough fruits to collectively and breath-of-a-hair-barely escaping the sinking of the entire island. These palms were fertile, I tell ya.

And there were them these pirates, ya know. Arrr! And they were five in all. Ok, you can stop arrr-ing now. Huh? What were these pirates doing on the island? Uh, they got stranded? How do I know? (*whispering to myself* the filthy swashbuckling buccaneers…)

Anyway, they saw them these coconuts just hanging there, and they thought they were in coconut heaven. They proceeded to gather all the coconuts they could get their hands on. Hmm? They might have some food on them. Uh, maybe they had cravings for coconut?

The monkey saw what they were doing, and went to help them. Finally, at sundown, they… what? It took maybe a day… Yes, that means the pirates arrr-ived in the morning. Alright, let me continue the story, will ya?

So at sundown, they were exhausted, and decided to divide the coconuts the next morning. And they went to bed. Or sand. Or whatever they used as makeshift beds.

During the night, one of the pirates woke up, fearing the others would betray him. And so he started dividing the pile of coconuts into 5. While he’s busy counting and dividing, the monkey came to the pile, took a coconut, opened it up and started eating. The fearful pirate, afraid of startling the monkey and making more noise, just let the monkey be. It turned out well, for he divided the pile evenly into 5. And then hid his share of the coconuts. And he went to sleep.

And another pirate, fearing the same thing, woke up just after the first pirate went to sleep. And he divided the rest of the coconuts into 5. And the monkey also took a coconut and ate it, bemused at the stealth tactics employed by the counting pirate. This second pirate was satisfied that he counted correctly and that he divided the pile evenly into 5, so he let the monkey go. Besides, he was busy hiding the coconuts and trying to be quiet.

A third pirate did the same thing, dividing the remaining pile of coconuts into 5. And the monkey did the same thing, eating a coconut while the pirate was having trouble because he didn’t have enough fingers on his left hand to help him.

And a fourth pirate did the same thing. And the fifth too.

At each quiet division of coconuts, the pile was divided evenly. And the monkey ate a coconut at each division. So, how many… Yes, the monkey ate a total of 5 coconuts. You’re so clever. So, how many coconuts were there in the original pile?

### The solution (sort of…)

I am going to work backwards.

Let y be the number of coconuts left over.
Let x be the original number of coconuts.
Note that x and y must be integers.

At 5th pirate division:
He hid his pile, so we add his pile back.
What’s left would be 4 parts for the rest, so we get 5y/4 to get all the pirates’ share.
Oh yeah, we need to get the monkey to regurgitate his coconut.
Ok, maybe not…

So before the 5th pirate divided, we had
5y/4 + 1
= (5y + 4)/4

At 4th pirate division:
It’s 4 parts for the other pirates, so we do the 5/4 multiple again to get
5( (5y + 4)/4 )/4
And we add 1 from the monkey to get
5( (5y + 4)/4 )/4 + 1
= (25y + 20)/16 + 1
= (25y + 36)/16

Note the recursive nature.

At 3rd pirate division:
5( (25y + 36)/16 )/4 + 1
= (125y + 180)/64 + 1
= (125y + 244)/64

At 2nd pirate division:
5( (125y + 244)/64 )/4 + 1
= (625y + 1220)/256 + 1
= (625y + 1476)/256

At 1st pirate division:
5( (625y + 1476)/256 )/4 + 1
= (3125y + 7380)/1024 + 1
= (3125y + 8404)/1024

And so we have
(3125y + 8404)/1024 = x
=> 1024x = 3125y + 8404

Now, my first thoughts had something to do with the Chinese Remainder Theorem or the GCD or something. And I searched for the solution, because I have no idea how to start. And I came upon Diophantine equations.

And I have absolutely no idea how to solve the equation.

Now my friend did tell me that 3121 was a solution for the number of original coconuts, which would mean 1020 coconuts were left over. And something about modular arithmetic.

What is more interesting, is that he told me -4 is also a solution. I plugged it into the equation, and lo and behold! There were -4 coconuts left over.

Wait, what? So there were -4 coconuts originally, and after all the sneaking and counting of the pirates, and the monkey eating it’s, ok, his, (presumably) non-existent -1 coconut, we have -4 coconuts left over?

The only reason why that solution was confounding us, ok, me, no wait, you, arrr, whatever, is that an assumption was flawed. We can’t have negative numbers of coconuts. Not in this reality dimension anyway. Arrr.

But it’s interesting. -4 is a stable solution, in chaos theory parlance. I mean parrrlance. Arrr.

Final thoughts: Why would each pirate fear that the others would betray him, but was honest enough to divide the coconuts evenly? And how would the first pirate divide 3121 coconuts into 5 while keeping quiet (that’s a lot of coconuts)? And keeping his head straight? And keeping the monkey quiet?

This is why, sometimes, just because you can solve a math problem, doesn’t mean you should. Because it might not make sense in the first place.

P.S. This was inspired by my friend who told me about the puzzle, and the International Talk Like A Pirate Day. Arrr.

[image by akurtz]