A story with hats

A king has just captured 3 female prisoners: Aline, Justine, and Corinne. He presents them six hats, three white ones and three black ones, and he tells them that he's going to put a hat from the set onto each of them, without unveiling its colour.

The king puts a white hat onto each prisoner. A few minutes later, Aline, Justine and Corinne are located in the main room of the palace, in a triangular configuration, such that each of them can see their friend's hats, but not their own. They know they're going to be asked the color of their hat, and that their answers will determine their fate. If a prisoner gives the right color, she'll be released, whereas in the case of a wrong answer, a special treatment, which won't be detailed here, will be reserved to her.

– Great, says the king, now who can tell me the colour of her hat?

Aline looks around her. She observes that both Corinne and Justine have a white hat. But that doesn't help her really. Ultimately nobody answers.

Q1 : show that whatever colours Justine's and Corinne's hats have, Aline can't guess her own.

– Well, says the king. In this case...

He's on the point to call his servant, then he seems to change his mind.

– OK, he says with a great smile. Let me give you a hint: at least one of you has a white hat!

"This is just wonderful", thinks Aline. "I already knew it. Moreover, Corinne too knew it since she can see Justine's hat, and Justine knew it as well since she can see Corinne's hat. What a sadist king."

– So? asks the king.

Aline, Justine and Corinne don't say a word.

– Come on, insists the king, no one?

The girls carry on keeping a low profile.

– Well, I'm afraid we don't have the whole day. You?
– But I have no idea, complains Aline. Besides, your hint is completely useless!
– If you take it this way... says the king.

Then he snaps his fingers and points the trio.

Q2: show that the prisoners could have freed themselves using the king's indication.


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