# Red or Black

There are 3 children sitting on three chairs. The children can only look forward and all wear one hat. The hats are either black or red. They are a total of 5 hats, 3 are colored red and 2 are colored black. The child on chair A says: "I don't know what color of the hat I have." Then the child on chair B says: "Also, I don't know the color of my hat." Then the child on chair C says: "In that case, I know the color of my hat."

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A        B        C


The question is: What color of hat does the child on chair C have and why?

Bonus question: Do we also know the color of the hat of the child on chair A and the color of the hat on chair B?

• @Gerhard was the edit necessary? It didn't really change anything except for the title and the spelling of "colour" vs. "color", which isn't that helpful. – mdc32 Feb 4 '15 at 13:50
• @Gamow But why change "colour" to "color"? That is your regional preference you've imposed on OP. Only commenting on this now because it somehow made it to the top page. – Trenin Dec 16 '15 at 15:20
• @Trenin, Nai answered on this question today. – martijnn2008 Dec 16 '15 at 16:10
• @martijnn2008 Ah! I see why. Still, valid comment though - why change colour to color? – Trenin Dec 16 '15 at 18:29
• yeah, I have no idea, but I don't care. – martijnn2008 Dec 16 '15 at 18:46

If A doesn't know the colour of his hat, then at least one of B and C must be wearing a red hat, because if they were both wearing black, then A would know immediately that he was wearing a red one.

B knows this as well. But if C were wearing a black hat, then B would know that she was wearing a red one because of that. So since B doesn't know, C must be wearing a red hat.

C knows this as well. So C knows that his hat is red.

As for the bonus question, no, we cannot know what colour hat either A or B is wearing. All else we can deduce from the problem is that at least one of B and C is wearing a red hat, but C wearing a red hat fulfills that already, so B could be wearing either colour of hat. And in general, nobody can know what A's hat's colour is, since nobody can see it. So all four arrangements where C wears a red hat are possible.

With three children, three red hats and two black hats, there are 7 different configurations possible:

$$\begin{array}{cccl} \text{Alice} & \text{Bob} & \text{Carol} \\ \\ \hline \text{red} & \text{red} & \fbox{red} & \text{(1)} \\ \text{black} & \text{red} & \fbox{red} & \text{(2)} \\ \text{red} & \text{black} & \fbox{red} & \text{(3)} \\ \text{red} & \fbox{red} & \fbox{black} & \text{(4)} \\ \fbox{red} & \fbox{black} & \fbox{black} & \text{(5)} \\ \text{black} & \fbox{red} & \fbox{black} & \text{(6)} \\ \text{black} & \text{black} & \fbox{red} & \text{(7)} \\ \end{array}$$

With knowable hats boxed.

If Alice sees two black hats (5), she'll know the black hats have been exhausted and she is wearing a red one. Since Alice knows the color of her hat, Bob and Carol will know that they are both wearing black hats, since this is the only configuration that allows Alice to know her hat's color.

If Alice sees at least one red hat, she doesn't know the color of her own hat and will state so. Now if Carol is wearing a black hat (4, 6), Bob will know the color of his hat, since Alice has seen at least one red hat and Carol's hat isn't it, so his must be. When Carol hears that Bob knows the color of his hat, she can deduce that her hat is black, or else Bob wouldn't have known.

If Carol is wearing a red hat (1, 2, 3, 7), Alice won't know the color of hers and neither will Bob know the color of his, because all he can see is that Carol is wearing a red hat, but he will not know whether Alice saw two red hats or just one.

As we can see, Carol will always know the color of her hat, while Alice will almost never know.

We can also see that guessing has a 50% chance of guessing right, when guessing only if the answer is not known.

Actually, that last bit about guessing is not true, due to the chances of the different combinations coming up.

The possibilities are :

1. RED RED RED : A won't know for sure, B know either B or C or both have RED but isn't sure, but C is sure he has RED since both declined.

2. RED BLACK BLACK : A knows for sure since rest have Black Hats. B knows too. So does C.

3. RED RED BLACK : A won't know for sure, B know either B or C or both have RED hence is sure since it sees a BLACK Hat ahead

4. RED BLACK RED : A won't know for sure, B know either B or C or both have RED but isn't sure, but C is sure

5. BLACK BLACK RED : A won't know for sure, B know either B or C or both have RED but isn't sure, but C is sure

6. BLACK RED BLACK : A won't know for sure, B know either B or C or both have RED hence is sure since it sees a BLACK Hat ahead

7. BLACK RED RED : A won't know for sure, B know either B or C or both have RED but isn't sure, but C is sure

Implies 1,4,5,7 are the possible cases/.

As A doesn't know which color his hat is, B & C's color are not the same. (means [B != Black] OR [C != Black])
If C's hat color [Black], according to A said, B would think his is [Red] definitely.
But B said he doesn't know; C's hat color is not [Black].

So the color of hat the child on chair C have is RED.