# Prisoners’ problem 6

Three prisoners meet a guard. The guard says: I have hats with me, two of which are black and the others are white. The prisoners ask: “How many hats there are all together?” He says: “It’s a secret!”. The guard immediately picks three of the hats and puts them on the prisoners’ heads. All prisoners see other’s hat colours but not their own. The guard asks them one after another: “what is the colour of your hat?”. The first prisoner replies: “I don’t know”. The second prisoner replies: “I don’t know either”. What does the third prisoner reply and what is his hat colour?

For details see this
This entry was posted in Combinatorics, Recreation, Wednesday Problem and tagged . Bookmark the permalink.

### 2 Responses to Prisoners’ problem 6

1. Rick Woods says:

Since the first prisoner says “I don’t know”, we know that at least one of the two other prisoners’ hats are white (otherwise the first would know his hat was white). We make the same deduction from the second prisoner’s perspective.

The second prisoner also heard the first prisoner’s answer before answering himself; therefore he knows that at least one of his own hat and the third prisoner’s hat is white.

Considering all the possibilities for the three colors of hats, the information in the above two paragraphs eliminates all those where the third prisoner’s hat is black. Since the third prisoner has all the information in the above two paragraphs, he answers “white”, and his hat is white.

2. Tuomas says:

Second and third prisoner’s hats aren’t both black, since first person didnt know the colour of his own hat.

Second prisoner sees the colour of the hat on third prisoner’s head. If it is black, then he knows hes own hat is white, because they both didn’t have black hats.

Therefore third prisoner knows he has to have white hat, since the second prisoner didn’t know the colour of his own hat.