Solution for #36

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.