**Please comment your solutions, questions and remarks.**.

There are infinitely many prisoners in a prison. The prisoners are taken to the yard and everyone is given a hat which is either black or white. No one knows the colour of his or her own hat, but sees the other’s hats. Then everyone have to guess his or her own hat’s colour. Those who guess correctly are given freedom. The prisoners knew about this event in advance and have had some time to agree on a strategy. How can they agree such that

(a) infinitely many of them get freedom? Level 1/5

(b) only finitely many of them stay in the prison? 4/5

The prison looked like this btw:

I’m confused. Do they make the guess at the same time? What kind of information can be passed to the other prisoners? If they don’t make the guess at the same time, is there a pre-announced order?

Ps. Still working on the previous one.

They make the guess at the same time, or at least they do not know what are the guesses of the others.

I wrote:

> The prisoners knew about this event in advance and have had some time to agree on a strategy.

This is the only information they can pass additionally to that they see each other’s hats. So no communication after they’ve got the hats.