
#1




Liars and Four Hats puzzle
4 men are lined up in alphabetical order each is wearing a hat of a different color, but none of them are wearing their own hat.
Alan Red Brad Blue Cory Green Don Yellow You are to ask them each two questions but the question must be about the color of their own hat for example you can ask Alan "Is your hat Red or Blue". Each of your questions must be about two hats, not just one. But you will not know if he is going to lie or tell the truth only that he will give one truthful answer and one untruthful answer to your 2 questions. For one of these questions each man is going to lie, and on the other question he is going to tell you the truth. Before asking the question you will not know if he is going to be lying or telling the truth that round. They may all tell the truth on the first question and lie on the second or any combination but they MUST lie on one question and tell the truth on another. You may only ask each man two questions about the owner ship of their hats. And you must assign the correct hat to each person based on the answers of your questions. Is there a solution ? and if so : Which two hats would you ask each man about ? Last edited by Brian; 06202012 at 09:35 PM. 
#2




Right??
I would ask Alan, is your hat green or yellow, twice. Once he lies once he doesn't.
I would ask Brad, is your hat yellow or red, twice. Once he lies once he doesn't. I would ask Cory, is your hat red or blue, twice. Once he lies once he doesn't. I would ask Don, is your hat green or blue, twice. Once he lies once he doesn't. So say first Alan says green then yellow. Brad says yellow then red. Cory says red then blue. Don says blue then green. With these answers (just as an example they all told the truth then lied.) I would compare the answers and see what the truth was, by the answers. This might not make sense to anyone, but I tried. 
#3




Would Alan be able to answer with "red," "blue," or "neither" in your example? Or is it a yes/no question?

#4




I don't think that would work out so well
What is Alan supposed to answer if his hat is Blue? He can't "tell the truth" and answer Yellow or Green.
I worked from the idea that the man's answer would have to be Yes or No  was his hat either of the two colors. However, I can't find a solution that works in all cases. The closest I come is asking each man about the three colors he isn't wearing, duplicating one between them. For example: Alan, is your hat blue or yellow? Alan, is your hat blue or green? If he answers Yes to both questions or No to both questions, I know his hat isn't blue. If he answers Yes to one and No to the other, his hat is blue  regardless of which was a lie and which was the truth. Using that methodology, I can work out the man to hat color in all scenarios but one  where every man answers the same thing to both questions. Either I'm missing a trick (a variation on the question that fits within the rules given  "the question must be about the color of their own hat" and "must be about two hats, not just one") or there isn't a solution that works 100% of the time. For mine, sometimes I can solve by the second question to the second man  and other times I get to the last man only to have two colors and two men that are not solved. 
#5




I am pretty sure this puzzle is solvable under a specific constraint and it is cleat that it's not explicitly given in the context of the problem's statement.
For example, when asked to choose between two colors offered, any person will either answer with a specific color or with "neither". When lying a person will always choose to answer with a concrete color rather than answering with "neither" (even though "neither" would also constitute a valid untruthful answer). The only occasions where you will be given "neither" as an answer are those when the person is telling the truth and is offered choices neither of which would make a truthful answer (such as asking Alan whether his hat was yellow or green when his hat is in fact blue for example; he would have to answer with "neither" when answering truthuflly). 
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