Three perfectly logical prisoners, Alice, Bob, and Carol are selected for a test by a warden.

The warden presents them with a box containing 5 hats: 3 red hats and 2 blue hats.

The prisoners close their eyes. The warden places one hat on each of their heads, chosen from the 5 available hats. The two leftover hats are hidden away. The prisoners are arranged in a line, single file, all facing the same direction.

Carol is at the back. She can see the hats on both Alice and Bob.
Bob is in the middle. He can see the hat on Alice.
Alice is at the front. She can see no one's hat.

All three prisoners know this setup. They all know that there are 3 red and 2 blue hats in total. And most importantly, they all know that everyone else is a perfect logician.

The warden addresses them all:
>If you can deduce the color of your own hat with 100% certainty, you may announce it and you will be freed. If you are wrong, you will be punished.

The warden first asks Carol (at the back): "Do you know the color of your hat?"
Carol looks at Bob's and Alice's hats and, after a moment, says: "I don't know."

The warden then asks Bob (in the middle): "Do you know the color of your hat?"
Bob, who has heard Carol's answer and can see Alice's hat, thinks for a moment and says: "I don't know."

Finally, the warden asks Alice (at the front): "Do you know the color of your hat?"
Alice, who has heard both previous answers but can see no hats, smiles and says: "Yes, I know my color. My hat is..."

>What color is Alice's hat, and what is the logical deduction that allows her to be 100% certain?

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Three perfectly logical prisoners, Alice, Bob, and Carol are selected for a test by a warden.The warden presents them with a box containing 5…

Three perfectly logical prisoners, Alice, Bob, and Carol are selected for a test by a warden.
The warden presents them with a box containing 5…