They can see others' hats. Who figures out their color, and how? From the free puzzle set on Think Fanny — next try Three switches, one bulb.
The puzzle
Three perfect logicians sit where each can see the other two. Hats are red or blue. They all know that at least one hat is red. In fact, all three hats are red.
They are asked, repeatedly, whether they know their own hat color. At first everyone says no. Then, using the others’ answers, they all become sure.
What color do they announce, and what is the core idea?
Answer
Each person who sees two red hats eventually deduces their own hat is red.
With the public fact that at least one hat is red, seeing two red hats plus the others' failed first announcements rules out the one-red and two-red cases. The only remaining consistent world is three red hats.
