How To Really Solve The Last Coconut Puzzle

The last coconut puzzle is a simple game that challenges your intuition around decision-making. I recently came across the work done by Leonardo Barichello, which inspired me to write about this puzzle.

It all begins with a hypothetical story. Imagine that you are travelling on a ship as part of your vacation. Due to an unfortunate accident, your ship sinks in the middle of the sea. Luckily, you survive the incident.

Even more luckily, you manage to make it ashore to an island. However, there is one problem. The island seems to be uninhabited and you are the lone survivor who has made it to this island.

After scavenging the entire island for something edible, you come across a delightful-looking coconut. Considering the hunger you are experiencing right now, this coconut might as well be a three-course meal with excellent drinks.

As you approach the coconut with great excitement, a monkey jumps down from a nearby tree. He tells you the following:

“Caution, stranger! This coconut belongs to me. If you really wish to have this coconut, then you will have to play a game with me and win.”

Naturally, you are startled and surprised. But considering the circumstances, you ask the monkey to tell you more, and he proceeds to explain the rules of the game to you.

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The Last Coconut Puzzle — The Rules

The monkey reveals two dice from behind his back. Then, he goes on to prove to you that these are both fair dice.

Once you are convinced, the monkey tells you that the game begins when he rolls both the dice at the same time.

After revealing this, the monkey goes on to explain the rules of the game:

1. If the largest number from both the dice facing up is 4 or lower, then you win.

2. If the largest number from both the dice facing up is 5 or 6, then I (the monkey) win.

3. If both the dice land with the same number on top, then we call it a draw.

4. We will decide who wins by considering the best of three outcomes.

5. If you win, you get to keep the coconut. If I win, you will become my slave.

Given the rather risky rules of this game, would you take this bet? Either way, what do you think your chances of winning are?

Spoiler Alert

Beyond this section, I will be discussing the solution to this puzzle. If you prefer solving this puzzle on your own first, then I suggest that you pause reading this essay at this point.

Once you are done with your attempt, you may continue reading and compare approaches.

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The Solution to the Last Coconut Puzzle

People typically tend to choose to the bet offer, as intuition says that the monkey has lower chances of winning this game. But because of the closed nature of this puzzle, it is quite easy to work out your exact chances of winning.

The first step is to create a mental picture of all the possible outcomes. Since we have 2 fair dice here, we have a total of 36 possible outcomes.

Out of these, considering rule 4 (“If both the dice land with the same number on top, then we call it a draw”), 6 scenarios result in a draw . The resulting scenario outcome table would look like this:

At this point, we could just brute-force the solution by marking which scenarios lead to your winning and which scenarios lead to the monkey’s winning.

Let you be ‘Player 1’ and the monkey be ‘Player 2’. For example, if the dice return (1, 4), then Player 1 wins. And if the dice return (1, 5), then Player 2 wins. The consequent game outcome table would look like this:

By simple counting, you can tell that you would win in 12/36 scenarios and the monkey would win in 18/36 scenarios.

We can also arrive at this result by considering the fact that the outcome of each die roll has an independent probability associated with it. As a result, rule 1 can be transformed as follows:

Probability of each die individually resulting in 1 or 2 or 3 or 4 = (4/6)*(4/6) = 16/36

By incorporating rule 4, your chances of winning is as follows:

[Probability of each die individually resulting in 1 or 2 or 3 or 4] − [Probability of both dice being 1 or 2 or 3 or 4]

= Probability of your winning = (16/36) − (4/36) = 12/36 = 1/3 = 33% (approximately)

Since this is a closed game, the independent probabilities of all possible outcomes need to sum up to 1. We can take advantage of this fact and compute the probability of the monkey’s winning as follows:

[Probability of any outcome] − [Probability of your winning] − [Probability of both dice featuring the same number]

= Probability of the monkey’s winning = (36/36) − (12/36) − (6/36) = 18/36 = ½ = 50%

As you can clearly see, the monkey has the upper hand in this game and is trying to lure you into enslavement!

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Final Comments

One of the common misconceptions about applying probability to real-world situations is to mistake probability for certainty.

In other words, even though you have a lower probability of winning in this game, it is not a certainty that you will lose if you choose to play this game.

That is precisely what makes this game random. For the probabilities to approach certainty, something known as the law of large numbers needs to come into effect.

Instead of a ‘best of three’ scenario, if the monkey invites you to play a ‘best of several million’ scenario, the probabilities would be much closer to certainties.

I have covered this topic in detail in my essay on how to really understand expected value. If you are interested in that discussion, check it out.

For now, I hope you enjoyed solving this puzzle. If you are interested in more puzzles like these, keep an eye on this space in the future!

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Further reading that might interest you: 

• The Strong Law Of Small Numbers
• The Bell Curve Performance Review System Is Actually Flawed
• How To Casually Guess Numbers After Dice Throws

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