Question

EXERCISE 7 Three little girls, Alice, Betsy, and Cindy were sitting in a classroom, each wearing a hat. Initially, no girl knew the color of her hat, but it was commonly known among them that each hat could be either white or red. Yet, in that particular instance, all their hats happened to be red. The teacher came into the room and asked if any giırl could identify the color of her hat. They all said "No"' as no one could see her own hat. The teacher then publicly announced that at least one of them was wearing a red hat. Note that this fact had already been known to each girl individually as they could have seen red hats of others. The teacher just made this fact commonly known. Then the teacher asked if anyone could identify the color of her hat now. Still, no one could. The teacher asked the second time if any girl could identify the color of her hat. Again, no one could. The teacher asked the third and last time if anyone could identify the color of her hat. This time, each little girl readily answered: "Yes, my hat is red!" Yes, my hat is red! Describe the reasoning that led each little girl to this conclusion

Answer 1

Before the teacher made the announcement the girls already knew that there were two red hats since one girl could see the red color hats of other two . After the announcement it became common knowledge .

When the first time teacher asked the girl 1 knew that her hat maybe white or red since the other two are definitely red . The silence of all the girls made each of the girl sure that there are 2 red hats in the room and their own hat is unknown .

When the teacher asked the second time , again 3 of them remained silent . So the 2nd girl now has come to know that even the 1st girl is confused about the color of her hat like she is . The 3rd girl also came to know that the other two are confused about their own hats only . Since the teacher announced that atleast one red hat which means all 3 could be red also . If they try to answer sequentially , then 1st girl knew by the silence of other two that others were also confused

The silence of each girls in 1st two instances tells that other two has red hats . So in the 3rd instance all of them could answer .