Yhyjyk

A warden lines up these prisoners. He announces, "I see a white hat." He then leaves the room. Every few minutes he comes back in and asks if anybody knows their hat color. Anyone who figured it out before the warden came back must announce "I do", after which he is freed. Everyone else must remain silent until the next visit. After a prisoner is freed, everyone else will know who was freed.

Assuming:

• each prisoner knows that there are 8 prisoners (including themselves), each with a hat, lined up in this orientation


• each prisoner can see the hat color of all the prisoners in front of them (not their own or those behind them)


• the prisoners cannot move or communicate at all, beyond announcing to the warden that they know their hat color


• each prisoner is a logician

Who (if any) figures out the color of their hat, and when?

Solution

This is a variation on the blue-eyed islanders puzzle, with the added condition that some people can't see others.

If prisoner 1 saw only black hats, he"d know that his own hat was white and announce that on the warden's first visit. Since he doesn't do this, all prisoners now know that the is at least one white hat among prisoners 2-8.

If prisoner 2 saw only black hats, and if prisoner 1 hadn't been freed on the first visit, then he'd announce that his hat was white on the warden's second visit. Since he doesn't do this, all prisoners now know that there's at least one white hat among prisoners 3-8.

Following this logic, on the seventh visit, all prisoners know that there's at least one white hat among prisoners 7-8. Prisoner 7 sees a black hat and knows that his own hat is white. From this prisoner 8 knows that his hat is black on the eighth visit.

The remaining prisoners will never have enough information to deduce their hat color. In general, the only prisoners who can deduce their hat color are the rightmost white-hatted one and the prisoners to his right.

(If the warden is feeling generous, he can say, "I still see a white hat" on each visit and all the prisoners will eventually be freed.)

Since the prisoners do not know how many hats of each color there are, none of them can determine their hat color. The first prisoner (at the right) has no information other than that he is wearing a hat, and that he's first in line. The second prisoner knows that there are people behind him, and that he sees a black hat, but he has no way of knowing whether he has a white or black hat, because the white hat that the warden sees could be behind him, or it could be his. The other prisoners all see both black and white hats, but since they don't know how many of each there are, none of them can be sure of the color of the hat they are wearing.

Because all of the prisoners are logicians, they deduce it best to let the last person in line answer. Initially, the last prisoner in line can deduce his hat color because he can see the other 7 prisoners and the orientation and color of the hats, so he is the first to leave. Sequentially, whomever becomes the last in line will then speak up when the warden returns. Eventually we're left with the first three prisoners in the original line-up because for the last in line the color is known but the orientation could be missed. Thus, each are left with a 50-50 chance of getting the color right. So they may play the odds and let the last person answer again and risk going free or staying in jail.

I can't comment yet so, is the assumption OP mentioned below the accepted answer:

The prisoners can trust the warden's statement to be true?