28 of 100: Hat Color Mystery

If both Jana and Karim wear a $\color{#E81990} \text{pink}$ hat, Iris will know that she's wearing a $\color{#20A900} \text{green}$ hat since there are only two $\color{#E81990} \text{pink}$ hats, a contradiction.

So, at least a person among Jana and Karim wears a $\color{#20A900} \text{green}$ hat. $(*)$

Furthermore, if Karim's wearing a $\color{#E81990} \text{pink}$ hat, combining with $(*)$ , Jana could know that she's wearing a $\color{#20A900} \text{green}$ hat, a contradiction.

Hence, Karim isn't wearing a $\color{#E81990} \text{pink}$ hat, so Karim's wearing a $\color{#20A900} \text{green}$ hat.

Suppose the girl on the left is Iris. The only way for Iris to know what color hat she has is if the other two have pink hats. Since she doesn't know, the other two can't both have pink hats. That gives six possibilities, as shown in the picture. Jana (middle) also knows that it is impossible for both she and Karim to have pink hats. If either of the middle scenarios happened, she would see Karim's pink hat and know that she was wearing a green hat. Those two scenarios are thus impossible. The only remaining scenarios have Karim wearing a green hat. Therefore, Karim is wearing a green hat.

The following image will express, each one how they are thinking in their turn, why they made the statement & the possible combinations.

Iris says, "I don't know what color hat I'm wearing."

This means that Jana and Karim cannot both be wearing pink, because if they were, Iris would know she had a green hat.

Jana then says, "I don't know what color hat I'm wearing."

Jana knows, from Iris' statement, that she and Karim are not both wearing pink

If Karim was wearing a pink hat, then Jana would therefore know that she wasn't wearing pink as well and was wearing green instead. But she doesn't know, so Karim is not wearing a pink hat.

Therefore Karim is wearing a green hat.

The people that speak in order are (I, J, K). There are 7 total possibilities: (G,G,G) / (G, G, P) / (G, P, G) / (P, G, G) / (G, P, P) / (P, G, P) / (P, P, G)

Iris doesn't know what she has, which means she didn't see (?, P, P), otherwise she would know she was G. So (G, P, P) is impossible.

Jana also can't deduce what she has, which means she didn't see (P, ?, P) because then she would know that she is G. So (P, G, P) is impossible.

With those two options gone, all other possibilities have Karim wearing G EXCEPT for the (G, G, P). Now is this case possible? If it was possible, then Iris would see (?, G, P) and not know what she is wearing. But that would tell Jana that Jana isn't wearing P (since if Jana was wearing P, then Iris would know she was G. But Iris already said she didn't know what she was.) So if (G,G,P) was the right answer, Jana would know that she was G after Iris didn't deduce what she was. But we know that Jana doesn't actually know what she is, so this case did NOT happen. Therefore, (G,G,P) is also impossible.

All other possibilities have (?,?,G), so Karim must be wearing G.

When Karim looks out at the other two players' hats, he will either see two greens, one green and one pink, or two pinks. If he sees two pink hats, he will know right away that he is wearing green.

If he sees a green and a pink hat, he will reason that if his hat were pink, either Iris or Jana would have seen two pink hats, and so would have known her own hat color. Therefore, he can again conclude that his own hat is green.

If Karim sees two green hats, he has to reason very carefully. He knows that if his hat were pink, Iris would have seen a pink hat and a green hat, and would not have known her own hat color. However, Jana would have seen Karim's pink hat and known that if her own hat were pink, Iris would have seen two pink hats and would have known her own hat color. Thus, Jana would have been able to conclude that her own hat was green. Since she was not able to do this, Karim once again knows that his own hat is green.

If Karim wears pink hat and sees at least one pink hat then she understands that either Iris or Jana would get who of then wears the green one. If Karim sees two green hats then she understands that Jana could get that she wears the green hat because otherwise Iris would see two pink hats and realizes that she (Iris) wears the green one.

So the only option is that Karim wears green hat

Iris and Jana must have green hats because they cannot tell their own hat colors given the others. But, given this information, Karim can determine that he is wearing a green hat (because his color does not allow the other two to determine their colors).

Suppose you're playing the game. There's only one way for you to know what color hat you're wearing: if you see the two other people wearing pink hats. Simply, there are no more pink hats for you to wear, so you must be wearing green. If you open your eyes and see one green hat and one pink hat, there are two green hats and one pink hat left, so you can be wearing either color. If you see two green hats , there's one green hat and two pink hats left, so you can be wearing either. Thus, you can assume that when Karim opened his eyes, he must have seen two pink hats.

From what Iris initially says, now we know that both Karim and Jana can't have a pink hat, otherwise she would know that she would have a green hat.

Jana knows this too. This means that if Karim was wearing a pink hat, then she would have a green hat for sure, and she would know it. However, she said she didn't know the color, so it is safe to say Karim is wearing a green hat.

Karim can obviously not wear a pink hat, because in that case one of the girls could tell, what hat she's wearing. Hence Karim's hat has to be green.

Karim must be wearing a green hat, which we can demonstrate by the following reasoning.

First, consider what Iris saw. She can't have seen 2 pink hats, otherwise she would have realized that the pink hats were used up, and she must have been wearing a green hat. But Iris didn't know what color hat she is wearing.

So Iris saw either 2 green hats, or 1 green and 1 pink hat.

Jana is smart, and so also came to this conclusion about what Iris saw.

Second, consider what Jana saw:

• If she saw Karim wearing a pink hat, she could have deduced the following: "Iris saw Karim wearing a pink hat, but she can't have seen 2 pink hats. Therefore I am wearing a green hat."

• But it didn't happen that way. Jana was unable to deduce what color hat she was wearing.

• Therefore, Jana saw that Karim is wearing a green hat.

Since the probability of wearing a green hat is 3/5 and the probability of a red hat is 2/5, it is more certain that he wears green. Actually using multiplication rule in Probability and conditional probability the same estimates results.

Best problem so far! I enjoyed it very much!

For Iris to say "I don't know what color hat I'm wearing", Jana and Karim either have (G, G), (G, P), (P, G) color hats respectively - and Jana now knows this. [If Jana and Karim both had Pink hats, Iris would know that his hat must be Green.]

• If Karim's hat was Pink, the above options show Jana could only have a Green hat.

• If Karim's hat was Green, the above options show that either Jana has a Green or a Pink hat.

Thus, since Jana isn't able to tell what color her hat is, Karim's hat must be Green.