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.