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# Another hat problem

## Question

Adam, Bill and Charlie stood in a line with Adam in front and Charlie at the rear. Charlie could thus see both Adam and Bill, Bill could see Adam, but Adam could not see either Bill or Charlie.Their teacher brought out a box containing three red hats and two blue hats. She placed on each boy's head a hat from the box and asked them to name the color of the hat they were wearing.

Although all three responded to the question, only Adam knew the color of his hat. And now so do you.

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Spoiler

Long story short: Adam must have a red hat on.

Let's start from Charlie's point of view:

• If Adam and Bill both had the blue hats, then Charlie would IMMEDIATELY say "RED!" (since there are only 2 blue available).
• If he does yell red immediately, then both Adam and Bill would know their hats are blue.
• Because the OP says that only Adam knew the color, this scenario can be dismissed.
• This means Charlie sees either one red/one blue hat or two red hats and therefore can't know what color his hat is.

Now we move to Bill:

• Because Charlie doesn't respond right away with "red", that tells Bill he and Adam couldn't both have blue hats.
• So if Bill sees a blue hat on Adam, Bill would immediately say "red"...
• If this happens, Adam should know based on the logic above that he has a blue hat on, and so he would say it
• This would mean both Adam and Bill know their color, so this scenario is out as well.
• So that means Bill must see a Red hat on Adam.

So based on the above reasoning, the fact that both Charlie and Bill don't "know" the color of their hats immediately, tells Adam that he MUST have a RED hat on...otherwise, at least one other person would know what color they have.

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