The blue-eyed islanders

Today’s Riddler problem is another classic. The current incarnation of the puzzle is about error-prone mathematicians, while the classic version is about blue-eyed islanders.

A university has 10 mathematicians, each one so proud that, if she learns that she made a mistake in a paper, no matter how long ago the mistake was made, she resigns the next Friday. To avoid resignations, when one of them detects a mistake in the work of another, she tells everyone else but doesn’t inform the mistake-maker. All of them have made mistakes, so each one thinks only she is perfect. One Wednesday, a super-mathematician, whom all respect and believe, comes to visit. She looks at all the papers and says: “Someone here has made a mistake.”

What happens then? Why?

Here is the solution:
[Show Solution]

2 thoughts on “The blue-eyed islanders”

  1. In order for the deeply nested meta-knowledge reasoning to happen it must be common knowledge that they all have perfect and deep reasoning skills. But clearly all of them have considerable doubts this is the case given all the paper mistakes they all know everyone else has made (I think it’s safe to say these aren’t spelling mistakes or typos but math related, or they wouldn’t feel so ashamed of it to the point of resigning).

    Since they all know at least the other 9 are flawed then none can be assured all the others will reason through to the bitter end because, even if none of them were to make a mistake in their own reaoning, they can’t be sure that the others would be sure of this. Even worse, since the visiting super-mathematician made it common knowledge that at least one is flawed, that means they all now know it’s guaranteed that none would be able to get past this uncertainty about the others.

    But this common knowledge isn’t even needed, just having mutual knowledge that a single one is flawed would be enough to guarantee that everyone’s nested logical chain of deductions will break at some point (they might believe perhaps just one does reason through it and just one might resign on the 10th Friday, they might think that but it wouldn’t happen, since all actually know 9 are flawed).

    I think all would realize that the induction would most likely break down on the 3rd or at most the 4th nested meta-knowledge level. They all know there’s at least 9 flawed, so they can’t be sure none will make a single mistake when trying to reach that minimum required 9th level of “I (mathematician A) know at least 9 flawed, and I know that B knows at least 8 flawed, and I know she knows that C knows at least 7 flawed, and those know D knows 6… [etc repeated for all 9] and furthermore all know that all will reason as such and know that they all know that… [etc] that all will reason through this correctly”. So, none resigns on any Friday. 🙂

    I know this is not in the spirit of the original blue-eyed islanders puzzle. But why would Oliver introduce precisely this type of uncertainty about their math/logic competency in this particular variation with far from perfect mathematicians, if it’s not precisely to call attention to this missing premise, that in order for the chain in meta-knowledge reasoning to occur, it must necessarily be also common knowledge that they’re all perfectly rational and flawless logicians? Indeed, the problem as stated everywhere else I’ve seen it includes such a statement “they all know they’re highly logical, and they know they all know this, and they know they know they know this, and so on”.

    But Oliver or the puzzle creator didn’t say any of it, and instead added language that introduces significant doubt about the reasoning skills of not just one (which would’ve been enough) but of all of these agents. Perhaps the omision was a lapse, but I’d prefer to think it was intentional (particularly given this one’s such a well known classic it might require a tricksy twist like this to keep puzzlers on their toes and not just answer out of memory, or worse, googling for the answer which would be quite silly).

  2. Or, if we do take for granted that it’s common knowledge for all of them that they’re all highly logical, I can think of another twist that prevents any and all resignations.

    If they actually care for each other’s well being (or at least their faculty’s since they in fact have a scheme to avoid resignations), then all one of them has to do is, right after the unfortunate announcement by their esteemed super-mathematician, just take out their phone and pretend to have been on a call which prevented them from hearing the announcement. Or pretend they were in the bathroom: “Oh, did I miss anything important?”

    The mathematician doing this knows it would then guarantee that none of the others can continue with the nested meta-knowledge induction all the way through, and all would know no other can either, so none would deduce their paper mistakes and once again none would resign on any Friday.

    Ah, but then each of them would arrive at this same ruse as the best solution to avoid resignations, and thus an awkward but quite funny situation would ensue in which several (all?) tried to claim they missed the announcement. Of course none would tell any of it to the ones who claimed not to have heard it, knowing that this would reinstate the fateful chain of reasoning that they each privately know would end in 9 or all resigning.

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