Three Prisoners Solution

So, as I said in hint two, there are two cases, let us consider them now. There are two cases since all prisoners raised their hands, you know this since you can see two prisoners and you know you yourself raised your hand.
Case 1.
Since all raise hands, we know either all are red, or two are red. For case 1, let us look at all red. In all red each person looks out and sees all red, they know that the other guys are raising their hands, and that is pretty much it. If we are thinking we are red, then if we put ourselves in the other guys shoes, they will be seeing us(red) and the other guy(red). We know the other guy is red since we can see him. So all the prisoners will be thinking the same thing, and seeing the same thing.
Case 2.
Now look at the case where we are blue and the other guys are red. Let us denote ourselves as A, and the other two guys as B and C.
A is blue
B is red
C is red
When we look out we see all red.
Now let us put ourselves in the shoes of B. He must be seeing one blue(A) and one red(C). What is he thinking…He sees one red and one blue, and of special interest he sees C raising his hand. In order for C to raise his hand, he must see a red, and since we(A) are blue in this scenario, C must be looking at B(red). So therefore B must conclude that he is red or else C could not be raising his hand.
Since we are the smart guy, A, we think out the above two arguments in our heads really fast and then realize that B is not guessing, therefore B must not know what he is. If case 2 was the situation we were in, then B should be answering, but since B is not answering, we must assume he is stuck thinking about case 1, in which case we can then say that since B has not answered, we must be red.

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