100 Monks

Hints and solutions available here.

In Normandy, there is monastery of 100 Trappist monks. They are highly logical and can determine the answer to any solvable mathematical problem. The monks also keep strictly to the rule of St Benedict. To discourage vanity, they have no mirrors or any other reflective surfaces. They remain completely silent and all times and indeed do not communicate with each other in any way at all. However, they always take meals together each day at breakfast.

On a routine medical visit, the local doctor informed them one afternoon that at least one of the monks had contracted a rare and fatal disease. All affected monks displayed the characteristic symptom of a red dot on their forehead. He told them that the illness had already passed the contagious stage so that no new monks would be affected. Most bizzarely, the disease works directly on the brain and kills those infected during the night on the day they realise that they are ill.

The Puzzle: Everything is fine until the morning of the 13th day after the announcement, when some of the monks do not turn up for breakfast (for they are dead). How many monks died?

15 Responses to “100 Monks”

  1. 12?

  2. Since the dead monks were discovered on the 13th morning, they would have died on the 12th night, so there should be only 12 dead monks.

  3. I see now you labelled a day with zero. In common usage, the “first day” would be the day with no predecessors. If by “n-th day”, you mean the day with n predecessors, then I think you should say so in the puzzle, not just the hints.

  4. Sorry, I’m a bit confused. Maybe someone can help. I can understand the example given if two monks are ill. Each knows that the doctor said ‘at least one’ would be ill, and they know that when a monk realises he is ill, he will die that night.

    Right, I’m okay so far. But when we start to make the assumption that this will work in a kind of compound way… I get a bit lost. Why would 13 monks end up assuming that each of them, as individuals, were ill, just because the other 12 hadn’t died yet? The realisation that you’re ill doesn’t tie in with a large number of others not having died yet… not beyond a pair of ill monks, does it? With a pair, there’s the logic that each of them can see that no other monks have the red spot on their forehead, so they assume they must have it. But why would all 13 eventually reach the realisation that they have the red dot, on the same (13th) day? They might suspect they have it, but how can they be sure?

    Sorry, I can’t quite grasp this. Can anyone help my soft squidgy brain, please?

  5. Laura — the working principle is that you can determine the solution to the n-monk problem if you know the solution to the (n-1)-monk problem.

    To see how this works, consider the case of three monks, Abelard, Bernard, and Charles (so n=3). On the day of the announcement, each of A, B, and C sees two dotted monks. Each monk immediately deduces that either there are two ill monks (in which case they themselves are safe), or there are three ill monks (in which case they are doomed to die).

    If Charles, for instance, were actually in the two monk situation, then he knows Abelard and Bernard won’t turn up for breakfast on the 2nd day. If they are munching down their porridge on the 2nd day, then Charles realises that all three of them have the disease. Abelard and Bernard apply the same logic, come to the same conclusion, and never make it for breakfast on the 3rd day.

    This gives the solution to the 3-monk problem, from which you can deduce the solution to the 4-monk problem, and so on.

    I hope this helps. Please do ask if anything is unclear.

  6. But if Abelard go to breakfast the first day and see 12 monks with red dots, and then next morning there is still 12 there. Why can they not understand that they are the 13th that night?

  7. Michael Jenkins on April 20th, 2009 at 1:52 pm

    There is a flaw in the problem. The problem states that all 100 monks eat breakfast together everyday. The problem also states that the disease had already passed the contagious stage when the doctor discovered it…which means that the disease was once contagious. Therefore, every monk would have gotten infected while eating breakfast with the others.

    The real answer is 100 monks died.

  8. Mahalo for your message, Michael. I see your point, but it doesn’t follow that all the monks would necessarily contract the disease even in the contagious phase. In the same way, one member of a family can contract a cold without the rest of the family falling ill. I hope you agree!

  9. By that logic, however, the rest of the problem fails as well. If we assume that not all monks contracted the disease, why should we assume that every monk sees every other monk every day, or bothers to keep track of how many dots they see, etc. By the same token, although they do not “communicate with each other in any way at all,” you’d think these logically-inclined monks would notice when everyone at breakfast is staring at their forehead.

    But that’s just me nitpicking. Fun riddle in any case.

  10. Further step, these monks know the catalyst for their demise is knowledge that they have a dot on their brows. The logical step is to remove the catalyst, and therefore choose to be ignorant, and not look at anyone’s faces for the next 100 or so days. Or in an effort to help others remain alive, wear a headband that would cover said dot, and all 100 get to live until they see a mirror.

  11. Brilliant solution Ajonymous! I like the headband idea to save everyone’s life.

  12. Multiplemicedice on May 2nd, 2014 at 2:24 am

    Interesting Puzzle!

    Let I be infected monk

    1I -> realizes he/she has the disease -> dies on 1st day
    2I -> realizes the other ‘first’ monks have not died on first day -> has the disease -> both die on second day
    3I -> induces from 2 -> still alive on second day -> dies on third day
    4I -> induces from 3 -> still alive on third day -> dies on fourth day

    13I -> induces from 12 -> still alive on twelve day -> dies on thirteenth day…

  13. Multiplemicedice on May 2nd, 2014 at 2:27 am

    @Ajonymous
    They are not allowed to communicate meaning they cannot relay a message to another person… Staring at one particular forehead for an extended amount of time is considered communication because a message is being relayed irregardless of whether it is consciously or unconsciously done.

  14. This only works for n=2…

    If n=3, Monk A knows that the doubt in the minds between B and C saves them from death. After all, he can see that they both have the spot, so even if he is lucky enough not too (which he hopes is the case), he knows that the doubt in the others’ minds will save them from death (they will never realise they are ill – never triggering their own death). All three monks come to the same conclusion and so all survive…

  15. Quizmaster – here is the flaw in logic:

    “…Each monk immediately deduces that either there are two ill monks (in which case they themselves are safe), or there are three ill monks (in which case they are doomed to die)…”

    They are not doomed to die by being ill and having the spot. They only die if and when they realise they have the spot. Monk A (or B or C) doesn’t know whether he is alive because he has no spot or because he doesn’t realise he has a spot. It doesn’t matter. The doubt saves them all…

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