Three logical people, A, B, and C, are wearing hats with positive integers painted on them. Each person sees the other two numbers, but not her own, and each person knows that the numbers are positive integers and that one of them is the sum of the other two.

They take turns (A, then B, then C, then A, etc.) in a contest to see who can be the first to determine her number.

During round one, A, B, and C pass. In round two, A and B again pass, at which point C states that she now knows all three numbers and that their sum is 144.

How did C figure this out?

Congratulations to pcbouhid for a real puzzler! I also started with the "their sum is 144" as a given, but on rereading decided we can not take that as a given (if all three, or even just one, was given that fact at the start, it would have been solved in ths first round). What they DO know at the start, is that there are three positive integers, and that the sum of the lowest two is the third: from this all would know that their own number was limited to one of two, either the sum or the difference of the other two. From that it follows (though is not stated) that no two numbers are the same, since we know that all have positive (not "non-negative" which would allow zero) numbers -- if you saw two identical numbers, you would know yours would be the sum, but no one on the first round deduces that (assuming, as we must, their being "logical" extends that far). (I trust we shall not find that the "solution" requires any such sexist premiss -- however, I do not see the word "guys" cited in a previous comment.)

I will admit to googling to see if there was a discussion of similar problems: some puzzles were similar, but not directly applicable here, so far as I could detect. We are given that there were five consecutive passes (each presumably logical given the facts), until C finally got inspired. The task question is very clear that we must find a set of inferences made by C (hence without antecedent knowledge of the grand total (unless C had previously encountered some analysis of the situation, which showed 144 was the only total which would work -- hardly a fair way to reach the solution, and in that case C would have known on the first round.)

It seems there must be some inference tree (a generalized approach to such puzzles, perhaps) which is opaque to me.