Exploring the green-eyed prisoner riddle

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Seemingly simple solutions often have important underlying concepts which we tend to skip.

There is a fun logic riddle involving prisoners on an island. I came across it few years ago. A somewhat modified version of it can be found in this Ted-Ed video:

This riddle came up in one of my recent conversations with a friend, and it soon blew up into an intense and mind-numbing logical debate. My friend seemed to think that there is a fundamental flaw in the entire explanation. The simplistic solution given in the video didn’t satisfy him. Before long, we were breaking down the problem into various scenarios.

According to him, the induction logic broke down when we transitioned from 3 to 4. His arguments were quite convincing (the details of which I will skip, since they were ultimately wrong). For a long time, we were stuck in a phase — I would demonstrate how the problem made perfect sense from one perspective, and then he would ask me to consider his method of deduction which made everything seem chaotic. Neither of us could readily figure out loopholes in the other’s logic.

However, once the dust settled and both of us were satisfied, I realized that the simplicity of the puzzle had few intricacies hidden under it. It demonstrates how we might not understand a problem fully even after solving it.

Some relevant questions should come to mind:

  1. The outsider is not providing any new information. What is her actual role in the problem?
  2. Is the transition from 3 to 4, 4 to 5 (and so on) really as intuitive as it seems?

In any case, pondering over the seemingly simple solution provides some nice insights.

3 Replies to “Exploring the green-eyed prisoner riddle”

  1. Interesting. It got me thinking and I realized that the outsider cheated by providing new information. I’ll explain how in the two prisoner situation with Bill and Adria. Before the outsider came in, this was the state of knowledge of each prisoner:

    Bill KNEW that at least one prisoner has green eyes, specifically, Adria. He can see her eye color.
    Adria KNEW that at least one prisoner has green eyes, specifically, Bill. She can also see his eye color.
    Bill DID NOT KNOW if Adria knows that ‘at least one prisoner has green eyes’, because BIll does not know whether he has green or red eyes.
    Adria DID NOT KNOW if Bill knows that ‘at least one prisoner has green eyes’, because Adria does not know whether she has green or red eyes.

    After the outsider’s visit, Bill now knows that Adria knows that ‘at least one prisoner has green eyes’. Same with Adria. This is indeed new information as this information was not available before.

    Therefore I conclude that the outsider was cheating.

    1. It is evident that the outsider brings some new information which helps the prisoners leave. Otherwise they would’ve left on their own.

      However, I think it is more than what you mention.

      Your logic certainly holds true in the 2 person scenario. In higher cases, everyone knows that the others know that ‘at least one prisoner has green eyes’. They start making up assumptions in their heads and reduce it to a 2 person scenario, where again they have new information like you pointed out.

      But is that all the information they need? No.

      They also need a concept of shared time. They need to know that everyone has started counting from exactly the same day. This is new information too. The outsider essentially introduces this shared clock and starts off a countdown from hundred.

      This puzzle wouldn’t have worked if the outsider had not given the information to everyone simultaneously.

      These two things weren’t addressed when I first encountered the problem and hence the complications. The Ted-Ed video does a better job at explaining the solution than my previous source.

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