Three highly intelligent sophomore women are trying out as cheerleaders. Their names are Alpha, Aplus and AOne. Part of the tryout is the following challenge:
The women sit in a triangle, each wearing a hat colored either red or blue. Each woman can see the other two women’s hats but not her own. The head cheerleader tells them that at least one hat is red. Each woman is to look at the other hats and say nothing unless she can deduce what color her own is, in which case she may say what that color is.
Alpha sits with her eyes closed. After a a few minutes of silence she announces, correctly, that her hat is red. How did she do it?
Solution
Alpha analyzed these seven configurations:
(Remember that at least one hat is red.) Alpha thought as follows. If the configuration was VI, then AOne would know immediately that her hat was red. But AOne is silent. Likewise, if the configuration were VII, then Aplus would be speaking. But she isn’t. This eliminates VI and VII.
Now if it were V, then Aplus would be watching AOne closely because if AOne’s hat were blue, then Aplus would know hers was red and shout it out. However, AOne is silent. This eliminates V. Similar logic eliminates all possibilities in which Alpha has a blue hat. Ergo, she has a red hat and can proclaim it.