The great town wizard needs a new apprentice, and has announced a competition for the post. Many applicants have tried, and only the three brightest remain for the final challenge.
The wizard gives each applicant a hat, and tells them that they have either a black or a white hat. The first one to figure out the color of her own hat and publicly announce it wins the challenge, and becomes the wizard’s apprentice. Also, if any of them sees another applicant having a white hat, they are supposed to say:
“I see someone with a white hat.”
However, the wizard, devious as she is, has given white hats to all three applicants. So, predictably, they all immediately announce that they see someone with a white hat, and then nothing happens for a while.
But, after a few minutes of silence, the most clever of the applicants announces: I have a white hat! She becomes the wizard’s apprentice, and everyone lives happily ever after.
How can she know the color of her hat?
Answer
We can figure out the answer by walking through the four possible cases: No white hats, one white hat, two white hats, or, the case that actually took place, three white hats. Lets call the apprentices Anya, Buffy and Cordelia.
If there are no white hats, then no one will say the’ve seen a white hat. Since they all said that they can see a white hat, this isn’t the case.
If there is only one white hat, given to Anya, then Buffy and Cordelia will immediately say that they see a white hat. Anya will know that it must be her hat that they see, and so he will know the color of her own hat. This doesn’t happen - they all see at least one white hat.
If there are two white hats, given to Anya and Buffy, they will all see at least one white hat, and say so. Cordelia won’t know anything more, but Anya and Buffy can reason further. Since Anya knows that Buffy sees a white hat, it must be her hat, since Cordelia’s hat is black. The same goes for Buffy. The quickest thinker of Anya and Buffy would claim to know their hat’s color almost immediately, and the competition would be over.
But this doesn’t happen - no one immediately wins. Since the winner would know her color almost immediately in all these cases, they must all have white hats. The quickest of them will know, and win.