A certain king wishes to test his three wise men. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then repeatedly asks them, ``Do you know the color of your spot?'' What do they answer?
The solution is that they answer, ``No,'' the first two times the question is asked and answer ``Yes'' thereafter.
This is a variant form of the puzzle. The traditional form is:
A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that his spot his white. How does he know?
The intended solution is that the wisest reasons that if his spot were black, the second would see a black and a white and would reason that if his spot were black, the third would have seen two black spots and reasoned from the king's announcement that his spot was white. This traditional version requires the wise men to reason about how fast their colleagues reason, and we don't wish to try to formalize this.
Here is the Mr. S and Mr. P puzzle:
Two numbers 
and 
are chosen such that
.
Mr. S is told their sum and Mr. P is told their product. The following
dialogue ensues:
Mr. P: I don't know the numbers.
Mr. S: I knew you didn't know. I don't know either.
Mr. P: Now I know the numbers.
Mr S: Now I know them too.
In view of the above dialogue, what are the numbers?
2007 note: At the time I wrote this article, I was unable to discover the author. It was Hans Freudenthal, Nieuw Archief Voor Wiskunde, Series 3, Volume 17, 1969, page 152).