The puzzle you propose can be solved by, say, drawing a diagram and mapping out the relatively limited set of possibilities. As the only variable is what #1 said, let’s consider what he said:

You ask him if he tells the truth. If he is a truth-teller, he’ll answer “Yes”. If he’s a liar, he’ll answer “Yes” also. So #1 will always answer yes to your question. Consider #2, then, who says that #1 answered “Yes”. #2 is a truth-teller, otherwise he would have said “No”. Lastly, consider #3 who says that #1 is a liar. If #1 is a liar, then #3 is a truth-teller. If #1 is a truth-teller, then #3 is a liar. So, while it’s not possible to determine whether #1 or #3 is the liar, we can say for sure that there are two truth-tellers and one liar in that group.

As noted, I feel this is less complicated because all the information is all there, and it’s just a matter of working through the possibilities. The original puzzle, on the other hand, required invention.

In a certain land, the inhabitants are only one type or the other. You meet three of them. You ask # 1 if he tels the truth. You don’t hear what he responds (“yes” or “no”). Number 2 says that # 1 responded, “Yes.” (“I tell the truth”). Number 3 says that # 1 is, in fact a liar. The question to solve is, “(based on just the above information), “How many of each are there?”

Email me if you think you’ve gotten the answer and explain what your reasoning was or if you want the correct answer and reasoning.