Question

Alphonse, Bertha, Conrad, and Dagmar are sitting blindfolded in a room when hats are placed on their heads. They know that each hat is either red or blue. In fact, each person was given a red hat. Once the blindfolds are removed, each person can see what colour hats the other people are wearing, but can’t see his or her own hat.

(a) Suppose that it is announced that there is at least one person wearing a red hat. First Alphonse, then Bertha, then Conrad, and finally Dagmar are asked what colour hat he or she is wearing. Everyone can hear each answer. How should each person respond?

(b) Suppose, instead, that it is announced that there are at least two people wearing red hats. Do any of the responses in (a) change?

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Answer #1

a) Alphonse, Bertha and Conrad would not know the colour. But Dagmar will know his hat is red.

Explanation - Alphonse knows there is atleast one red hat. But he see that all his 3 friends are wearing red hats. So he cannot deduct the color of his hat.

But this also gives a valuable information for the other 3 players. If Bertha, Conrad and Dagmer were wearing blue hats, Alphonse would have known that his hat is red. But since Alphonse doesn't deduct his hat color, all other 3 players can infer that atleast one of them wearing red hat.

Now, it is Bertha's turn. Bertha know atleast one of Bertha, Conrad and Dagmar is wearing red. But Bertha can see red hat on Conrad and Dagmar. So Bertha cannot infer own color. But now, Conrad and Dagmer know one of them is wearing red. But Conrad can see red on Dagmar. So Conrad also cannot infer his hats color. Since Conrad also cannot infer the hat color, Dagmar can infer he is wearing red.

b) Conrad and Dagmar will now infer they are wearing red.

Explanation- Alphonse cannot decide his hat color. So other 3 players know atleast 2 of them are wearing red. Bertha see 2 red hats. So Bertha also cannot infer her hat colour. Now, Conrad and Dagmar will know they are wearing red.

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