### Grant McDermott

Assistant Professor
Dept. of Economics
University of Oregon

# A riddle: Hats and signalling

Some of you may know this one already, but a here's a favourite riddle of mine.

Four prisoners are blindfolded and arranged as follows: Three of them are lined up one behind the other (i.e. facing the same way), while the last prisoner is placed on the other side of a screen. A hat is then placed on each of their heads. When the blindfolds are removed...
• Prisoner A can see B and C in front of him, but not D.
• Prisoner B can see C, but neither A or D.
• Prisoners C and D can only see the screen.
• None of them is able to see what colour hat they are wearing themselves.

The prison warden then announces his beastly scheme: "Each of you has either a red or a blue hat on your head. There are two hats of each colour. One of you will be able to say $$-$$ with 99% certainty $$-$$ what colour hat you are wearing. Moreover, you should be able to do so within a minute. Should that person call out the correct answer, you are all free to go. Failing that, you are all to be sentenced to death!"

[Cue: Dramatic music]

Which prisoner calls out the correct answer and how is he able to do so without resorting to a 50/50 guess? (Rest assured, this a puzzle of logic and so there are no petty tricks like whispering to each other, turning around, mirrors, peaking over the screen, etc.)

Solution below the fold!

===

After a few moments of tense silence, Prisoner B correctly calls out, "I'm wearing a red hat!" And, with that, the warden is grudgingly forced to let the men go free.

The trick to solving the puzzle is to realise that a different set-up of hats would make it very easy for someone to give the answer. In particular, Prisoner A would instantly be able to call out the right answer... if only B and C were wearing the same colour hats. However, since they aren't, he is forced to remain silent.

Nevertheless, Prisoner B realises this and $$-$$ given A's extended silence $$-$$ he deduces that he (i.e. B) must be wearing a different colour hat to C. Thus, he simply looks at the colour of C's hat and calls out the opposite!

In other words, it is A's silence that provides the crucial clue to B.

THOUGHT FOR THE DAY: I thought of this when I saw a Modelled Behaviour tweet yesterday: "Not signalling can be the ultimate signal".