Hats logic

We gather that from the 3rd person (who can see everything) saying he doesn't know that the first and second DO NOT both have white hats.

If the 1st person was wearing a white hat, the second person would immediately know his was black since they can't both be white as the third person does not know his colour.

However, since the second person does not know his colour, we know that the first person is wearing a ${BLACK}$ hat

If the last person sees the first two are both wearing white hats, then he would know he is wearing black. Since he says no, then the first two are not both wearing white. They are wearing either white-black or black-black.

If the second person sees the first person is wearing a white hat, then he would know he is wearing a black hat. But he says no, this means that the first person is wearing a $\boxed{Black}$ hat.

I was the one with black hat

The only time the last one will know what color he wears is if the 2 in front of him are both $\space\textbf {white}\space$

The only time the middle one will know his hat color is if the one in front of him is wearing a $\space\textbf { white}\space$ one also ... by combining the answer of the last one he will know that the last didn't know because he was seeing 1 white and 1 black

The first one will then know that his hat is $\space\textbf {black}\space$ since the middle one couldn't know

The person in the back says "I don't know" which means the two people in front of him don't have both white and white hats (otherwise he would know his color because there are only two whites), they either have black and black or white and black.

The middle person also says "I don't know", if the person in the front had a white hat, he would know he had a black one (He can't have white in this situation because then the person in the back would know his color) , because from what we figured out before the combinations for the front and middle person are white and black or black and black. Because the person in the middle says "I don't know" we know that therefore the person in the front has a black hat.

The wording is a bit confusing. But, I assume that the one who knows what color hat he is wearing can see that the other 2 are wearing white hats and knows that there are only 2 white hats. He thus can deduce that he must have a black hat because all of the white hats are taken... I guess.

We have 3 people, let's name them 1(front),2(middle man), 3(last man who can see everything)

There are three black hats and two white hats. If 1 & 2 both had white hat then 3 will know for sure he has black hat since there is no more white hat.

Now there are 3 situations A. 1 and 2 have black hat B. 1 has black hat and 2 has white hat C. 1 has white hat and 2 has black hat

We know 1 and 2 both cannot have white hat since if it were so 3 would know his hat colour. 2 know this fact too. If 1 has white hat then 2 know for sure he has black. Since black doesn't know his hat colour 1 definitely has black hat.

We can assume that the 2nd and 3rd both wear white hats then obviously the 1rst one will only wear a black hat hence he knows the colour of his hat

For the 1st guy to know his hat colour there needs to be only one possible hat colour left after the other 2 have chosen. This can only be done if the other 2 choose white hats and he chooses a ${Black}$ hat