The Three Wise Men Puzzle: Answer

This is the answer to the "Three Wise Men Puzzle" posted earlier.

To understand the reason the men stop laughing we follow the thoughts of one of the philosophers, let's call him A. A sees the other two men, B and C laughing. Assuming that he doesn't have a mark on his forehead, he thinks that B is laughing at C and vice versa. But he must also assume that B is unaware of his own mark on the forehead. If A had no mark, what does B think C is laughing about? In other words, if A's assumption was true, B should quickly realise that C is laughing at B and therefore B should stop laughing. But after enough time has passed, and B hasn't stopped, A must assume that he himself has a spot on his forehead.

Three Philosophers Puzzle (Three Wise Men Puzzle)

Here is a logic problem which I was told some years ago and I found it quite interesting at the time. I tried finding a reference to this problem on the internet but wasn't really successful.

EDIT: Thanks to one of the readers, who pointed out that the puzzle is actually well known and is called the "Three Wise Men Puzzle". A variation of the puzzle is the "Muddy Children Problem".

The problem goes like this:

The Three Philosophers  -  by Giorgione

Three philosophers are sitting in the shade of a tree discussing deep philosophical problems. They talk and talk and, of course, they don't reach an agreement or come any closer to a solution. After some hours of talking the afternoon sun makes the three men grow tired and they finally decide to take a nap. While the philosophers are sleeping a young boy from the local village passes by. He looks at the men and decides to play a prank on them. The boy takes some white paint and on each of their foreheads he paints a white spot that resembles bird droppings. After some time the philosophers wake up again and, looking at the others, they all burst out into laughter. Of course each of them thinks that a bird has relieved itself on their colleagues foreheads but none of them is aware that, he too, has a white spot on their forehead. In this way they keep on laughing, but they don't let the others know of their plight. But suddenly they all stop laughing, suddenly realising that they too must have the mark on their forehead.

Question: How, just by using logic, did they arrive at this conclusion?

The answer can be found here.