Riddle Of The Hats

Imagine a windowless room with no mirrors or reflective substances of any kind. Just four walls of drywall. In the middle of the room are three desks set up one behind the other in single file. There is a man sitting at each desk. For reference purposes, we’ll label the men, from front to back, A, B, and C. Now, by the way they are positioned in the room, man A can’t see anybody, man B can see A but not C, and man C can see both A and B. Now…

…a fourth man walks into the room. He reaches in his bag presents the men with five hats, three of the hats are red, two of them are white. He tells the men,

“I’m going to blindfold all three of you, put one of these five hats on each of your heads, and then hide the remaining two hats in my bag.”

So he proceeds to do just that — he takes out three blindfolds, blindfolds all three men, puts one of the five hats on each of their heads, and then hides the remaining two hats. He then says to the men,

“I am going to remove your blindfolds. When I do, you are not allowed to turn around or talk to each other.”

He then removes the blindfolds and says,

“The first man that can stand up and tell me what color hat he has on his OWN head will get a million dollars. If you stand up and guess wrong, I’ll cut your head off. In other words, no guessing.”

After a few minutes of silence, man A stood up and stated what color hat he was wearing. He was correct.

What color hat was man A wearing, and how did he know?

• jaced June 1, 2005, 7:27 pm

There are several variations to this riddle, all exercising the same basic logic.

One variation worth noting:

Three mathematicians come to an island. They meet a tribe of cannibals, who mean to eat them. However, being the sporting kind, the cannibals (as in the variation above) set the three mathematicans up in single file, take out the five hats (three red, two white), blindfold the mathematicians, put a hat on each of their heads, and hide the remaining two hats. They then say that any man that can state the color of their own hat will be set free.

The question is: How many mathematicians have a chance to live?

• andreas June 1, 2005, 7:37 pm

manA = red
the other two would only be guessing. (thankfully, they didn’t)

• Cindy July 11, 2006, 5:53 am

This riddle is driving me crazy … can you please post the answer? Or maybe it is posted, but I don’t know where to find it.

• Brian March 11, 2015, 1:16 pm

Imagine the perspective of man C. If man A and man B were both wearing white hats then man C would quickly know that his hat was red. Since man C did not quickly draw this conclusion this means that man A and man B are wearing different color hats.

Now what does man B see? If he looks ahead and sees a white hat on man A’s head then he must assume that his hat is red, for if it were white then man C would have quickly won the 1,000,000 bucks. But man B cannot answer with complete confidence so this means that man A’s hat is red.

So, man A waits a few minutes and realizes that because man B and man C are unable to ascertain the color of their own hats this means his own hat is red, for any situation in which his hat would be white the other men would have been able to quickly anwer. Hence his hat is red.