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.

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?