The logic is done piecemeal and you're assuming all the guys are clever enough to understand the situation.
Let's start with the last guy (the one facing the other two) since he has the most information of them all. If anyone has a chance at a first guess, it should be him.
Oh, but wait - he's not saying anything. Why not? Obviously, it's because he can't make a conclusion. However, if the front two guys were wearing black hats, then he could make a conclusion, because there are only two black hats. His hat would have been white, in this case, and he could leave. But he's still here, so we can safely conclude that at least one of the other two guys has a white hat. This is our first clue.
Okay, what about the second guy? Why doesn't he say something? Well, he knows that either himself or the guy in front of him is wearing a white hat - but what are we saying? He can see what hat the guy in front has! If it were black, he'd know that he was wearing the white one and he would leave. But again, he says nothing so the guy in front of him can't be wearing the black hat.
Therefore, the guy in front knows he's wearing the white hat...
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