Lateral Thinking X

[Holmes]

My turn again!

Ladies and Gentlemen, "Lateral Thinking X"

Four intellectuals are lined up so that each intellectual can see the ones in front of him but not the ones behind him. (The back one can see the other three, and the front one can't see anybody.) One hat is placed on the heads of each of the intellectuals. (None of them may see the color of their own hat, but each may see the color of the hats on the intellectuals in front of him.) Each of the four hats are one of three different colors (red, white, and blue), and there is at least one hat of each color (so there's one duplicate). Each of the intellectuals, starting with the back and ending with the front, is asked the color of the hat he is wearing. Each of the intellectuals is able to deduce and give a correct answer out loud, in turn. What arrangement of the hats permits this to be possible without guessing (since the specific colors chosen are arbitrary, just indicate which two intellectuals must be wearing hats of the same color), and how did they do it?

[S.H.]

Do the intellectuals know about there are 3 different color hats with 1 duplicate? If yes, then:

Spoiler:

Assuming red is duplicate..... the red and duplicate should be at the front.

Spoiler:

Lets assume the intellectuals by A B C D from back to front...For A to know his color, he should not be one of the 2 men with same same hat color(Assume he is blue)  After giving the answer OUTLOUD, the B will know what his own color is since his color will be different from the 2 Red in front of him and Blue behind him, which is white. Then as for C and D, since A and B answered correctly, then it only means that C and D are duplicates, because if not, A and B wont know their own hat color.

[tera]

Would it be...

Spoiler:

The front two have the same color and each person calls out his hat color from back to front.

If the duplicate is farther back in the line, the deduction will get "stuck" at that point in the line; therefore when it comes to the front of the line, since everyone else has been able to figure out their color without guessing, the front person knows that their color is the same as the person behind them.

[sstimson]

My turn again!

Ladies and Gentlemen, "Lateral Thinking X"

Four intellectuals are lined up so that each intellectual can see the ones in front of him but not the ones behind him. (The back one can see the other three, and the front one can't see anybody.) One hat is placed on the heads of each of the intellectuals. (None of them may see the color of their own hat, but each may see the color of the hats on the intellectuals in front of him.) Each of the four hats are one of three different colors (red, white, and blue), and there is at least one hat of each color (so there's one duplicate). Each of the intellectuals, starting with the back and ending with the front, is asked the color of the hat he is wearing. Each of the intellectuals is able to deduce and give a correct answer out loud, in turn. What arrangement of the hats permits this to be possible without guessing (since the specific colors chosen are arbitrary, just indicate which two intellectuals must be wearing hats of the same color), and how did they do it?

This must be my day for puzzles.The answer

Spoiler:

First should be the line up 1 in front 2 behind and 1 behind them
Next the back see all the hats but his so his is one of a kind say white.
The guy in front see no hats so his is also one of a kind say red
that leaves the two in the middle say blue

How it works

Back guy see three hats in my case two blue and a red So he knows his hat is white

The two middle guys can not see the back hat but they can see their friend next to them and the front guy. Since the Back Guy say white they know his hat they each see the red hat in front and being lateral thinkers know his hat is one of a kind , they also know that the guy's hat in back is also one of a kind for the same reason so they both know what the guy to the side is wearing is what they are wearing and in this case it is blue

The guy is front know the back guy is white, and the middle guys is blue so he knows his must be red

Later

[Holmes]

@ everyone:

Spoiler:

Answers all correct! They are very similar.
The two intellectuals in front must be wearing hats of the same color. Let's suppose the front two were wearing red hats, the third was wearing a white hat, and the fourth (in back) was wearing a blue hat. The intellectual in back must be the first to answer. If he saw one hat of each color on the three intellectuals in front of him, he would not be able to guess the color of his own hat, since the duplicate color could be any of them. Therefore, he must see two hats of the same color (red) and one hat of a second color (white), and he can state conclusively that he must be wearing the hat of the third color (blue).

Since the back person can say what color hat he's wearing, the other intellectuals must realize that no one else is wearing a hat of that color. So each of the others can narrow down the color of their own hats to the remaining two colors.

The next intellectual knows his hat isn't blue, and he knows there is only one hat that's blue. If he saw a hat of each of the two remaining colors on the two intellectuals in front of him, he wouldn't be able to determine the color of his own hat, since the duplicate color could be either of them. He must, therefore, see two hats of the same color (red), and can conclude that his own hat is of the color he does not see (white).

The next intellectual realizes that the only way the two intellectuals behind him could guess the colors of their hats would be if he and the front intellectual were wearing hats of the same color. He sees the color of the front intellectual's hat (red) and states that this is the color of his.

The front intellectual realizes this too and repeats the color stated by the intellectual behind him.

My answer 

[bash7353]

Okay, here's what I think:

Spoiler:

To make this simpler the red hat is thereafter R, the white one is W, and the blue one is B.
If the man being at the end of the line sees R, W, B, he can't conclude which color his hat has, because he doesn't know which color is duplicated and which aren't. So one color belonging to the first three people must be there two times. Which color that is doesn't matter, what matters is at what position these two same colors are. Let's assume the first three people's hats' colors are RRW (It's not clear yet at what position these colors are.). The last person can now deduce his hat is B.
In this case there are two possibilities as to what colors the 3rd in the line can see: RW or RR. Additional he knows the last person's color is B.
Say he sees RW: He can't know what his color is, the only thing he can conclude is it's not B, because then the 4th in the line would not have been able to make out his color. But he doesn't know whether his hat is R or W, so that is not an option.
Say he sees RR: Now he can see, that one man is B, two are R, so he must be W.
Now we come to the second in the line, he sees R in front of him, knows about W and B behind him. The only way for him to know he's R is: Otherwise the two people behind him would not have been able to determine their colors. Just like the 2rd, the 1st in the line can also come to the conclusion that the colors in the front have to be same and say his hat's also R.