## Circular Prisoners

A jailer takes 4 prisoners out to the yard to play a deranged game.

The jailer arranges all 4 prisoners in a circle, facing inwards, and then blindfolds each of them. Next, he places a hat - either red or blue - on each head. Then he removes the blindfolds. All prisoners can see each other’s hat color, but not their own. There is no form of communication allowed.

The jailer then shouts go. At that point, each prisoner may immediately guess the color of his or her hat, or say nothing. This is done simultaneously, so prisoners cannot wait to hear what another has to say before making his or her decison.

All 4 prisoners are set free as long as no one is wrong and at least one person is right. For example:

• If I guess correctly and nobody else guesses, we all go free.
• If all of you guess correctly and I guess incorrectly, none of us go free.

As usual, the prisoners are told the rules of the game beforehand and are given a chance to come up with their best strategy beforehand. What’s their best strategy and what’s the chance that they all go free?

Solution