Solution #18: Colored Labels

1. If Alice sees two blue labels she will know that her label is red and will declare this and since a little time has passed without this happening Betty and Carol cannot both have blue labels.
2. Similarly, if Betty sees two blue labels he will know that his label is red and will declare this, and hence, Alice and Carol cannot both have blue labels.
3. Carol can now deduce that if her label is blue, both Alice and Betty must have red labels. Now Carol knows that both Alice and Betty are intelligent. If Betty and seen a red label on Alice's hat and a blue label on Carol's hat, she would know her hat did not have a blue label, because in this case Alice would have declared her label to be red.
4. Since a little time has passed and neither Alice nor Betty has declared she could deduce the color of her label, Carol knows her label cannot be blue, and therefore must be red.