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?
12?
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.
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.
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?
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.