Does everyone have to answer? If so, do the others hear those answers?

These criminals want their freedom, so you can assume that if they are able to logically solve the king's problem, they will definitely shout out their answer.

The answer does involve another criminal's "guess". You can just assume this from the problem.

Well, if the person that goes free depends on another's answer, it sure as hell isn't the guy who yells out the answer first. So it's either the second or the third guy to answer that goes free.

Well, let's try this...

The first guy to shout out his answer can guess either triangle or rectangle. If he sees 1 guy with one and 1 guy with the other, obviously, he's going to shout out the rectangle because there's a greater chance for that one. If he sees two triangles, then, of course, he's going to shout rectangle because there are only two triangles, and he'd be right, but that's not the case. If he sees 2 rectangles, then he's going to go for the triangle, because then he has more of a chance. Regardless of what he shouts out, he's wrong. So the First guy can ONLY see either 1 of each, or 2 rectangles.

The second guy takes the first guy's answer and thinks about it. If the first guy is definitely wrong, then it's impossible for both the second and the third guy to have triangular hats. Because then the first guy would be right. Therefore, one of the remaining two must have a rectangular hat.

In essence....

If the first guy's answer is Rectangle, then the second guy knows that the first guy saw 1 of each. He sure as hell didn't see 2 triangles, because he'd be right. He sure as hell didn't see 2 rectangles, because he'd be gambling on a lesser chance.

In this case, the second guy looks at the third guy and guesses the opposite of what the third guy has.

****

If the first guy's answer is Triangle, Neither one of the remaining two must have triangles because he wouldn't gamble it if they did. Therefore, the remaining two have rectangular hats.

In this case, the second guy can safely guess Rectangle.

So I guess the second guy goes free then.

Well, let's see, I assume the criminals don't know that.

If the first guy says Rectangle, the second guy will still get it right.

I don't know if he says Triangle though....

So There are 5 rectangles or 3 rectangles?

Well, there's no way in hell the first person to guess gets it right if the answer depends on that of another person. Did the guy who got it right do something unorthodox? Did he have any form of help that the other criminals didn't get? Is this just logic or is there a catch?

Man C guessed rectangle.

Man A is rectangle
Man B is triangle
Man C is rectangle

Man A sees Man B with triangle, but sees Man B with rectangle. He guesses triangle because he knows there are two of them. This lets man C know he can't possibly be triangle.

Man C guesses rectangle, because he knows if he was a triangle, man A would have guessed that with both triangles out, he must be a rectangle. Thus, Man C has to have rectangle.

The criminal with the triangular hat is able to say he's got a triangle hat. Simply by knowing if the other two criminals have the same hats from his own observation, then the king must have "selected" him to be the one forgiven. To have given them all rectangular hats would mean they king wouldn't want any of them freed/forgiven.

The criminal with the Rectangle hat is freed, since he saw there were only two triangle hats, and each of the other criminals was wearing one, so his HAD to be rectangular.

OK, last time I had the number of rectangular and triangular hats reversed. This answer might be a bit of a loophole, but it's worth a shot. The criminals can't touch their own hats, but can touch each other's. One of the guys with a rectangle on his head notices when the other rectangle guy reaches out to the edge of the first guy's hat, he reaches out further to the side than he would have to if he was touching the outer edge of the triangular hat.Therefore he's able to deduce he has a rectangle on his head.

Think about it. It's Passover and the king has to free a prisoner.

Did you mean Man A sees Man B with triangle, but sees "Man C" with rectangle? He wouldn't guess triangle, because if he sees a rectangle and a triangle, he has no way of knowing whether he is triangle or rectangle. You're close; try thinking of another solution that does not involve probability. You're assuming that Man A is kind of sacrificing himself for Man C. Remember, these criminals want their OWN freedom. They wouldn't help another criminal.

This is a logic problem. Although the answer, as I said in a previous hint, depends on another criminal's guess, I didn't say if that criminal would verbalize his guess or not. Consider this.

Let's call the criminals Ecks, Wye, and Zee.

First up to guess is Ecks. If he saw two triangular hats, he could've confidently claimed his hat was rectangular (since there are only two triangular hats). If he couldn't deduce, this means that he saw at least one rectangular hat on Wye or Zee.

If Ecks is wrong/quiet, then we come to the Wye's guess now. Again, if Wye saw two triangular hats, he could confidently say his hat was rectangular. But if Wye sees a triangular hat on Zee, Wye could confidently say his hat was rectangular because he could deduce that Ecks saw at least one rectangular hat. However, if Wye can't figure it out, it's because Zee is wearing a rectangular hat.

So... if it gets to Zee because of the silence/wrong guesses of Ecks and Wye, Zee can say he's wearing a rectangular hat because Ecks saw at least one rectangular hat (on Wye or Zee) and because Wye did not see a triangular hat on Zee.

Game done.

Given:

Persons A, B, and C

A = Rectangular Hat (A is smarter than B)
B = Rectangular Hat
C = Triangular Hat

A, B, and C know that there are 3 rectangular hats and 2 triangular hats. They can only see the hats of other people.

Solution:

Person A tells the King that he has a rectangular hat, because seeing that C has a triangular hat, he reasons that if he also had a triangular hat, Person B would have shouted out the answer (that he has a rectangular hat).

Person B could have also reached the answer in the same way Person A did, but because the question specifies that Person A is smarter, Person A gets it first.

Person C does not have enough information to determine his own hat.

*Note, the key in this problem is recognizing that there are THREE rectangular hats and TWO triangular hats, and THREE people.