$\textbf{Toggler's Problem}$
Among a group of $100$ people, only one is a truth teller and the rest $99$ are togglers. A truth teller always tells the truth. A toggler will tell the truth and a lie in an alternating fashion. That is, after he or she tells the truth the first time, this person will tell a lie next time. However, if his or her first answer is false, then the next answer will be true. It is unknown whether a toggler's first answer is the truth or a lie.
If all these people know who is the truth teller, how many questions do you need to ask in order to identify the truth teller?
$\textbf{Solution}$
Two questions will be sufficient to single out the truth teller. Pick up anyone and ask these two questions:
If the answer to the question 1 is "yes", then ask
If the answer to the question 1 is "no", then ask
The answer to the question 2 points to the truth teller.
$\textbf{Analysis}$
Clearly, the key to solve this problem is to be certain whether the person's statement is the truth or a lie. If we know the answer is going to be true, then we can ask directly, i.e. who is the truth teller. If we are sure that the answer is going to be a lie, then we can ask a negative question: i.e. who is NOT the truth teller. Remember, double negative is positive. Therefore, we can always get a positive result as long as we know whether the answer is true or not.
Because there is no way to know the answer to the first question is true or false, therefore it is impossible to assert the result by just asking one question. This means if asking two questions can lead us to the result, it will be the best case. In order to achieve this optimal result, the goal of our first question is to ensure we know whether the second answer will be true or false.
If the first answer is "yes", his or her second answer must be true. This is because
Therefore, as long as the first answer is "yes", we can be assured that the second answer must be true.
Otherwise, if the first answer is "no", then this person cannot be a truth teller and must be a toggler. Furthermore, we know this answer itself is a truth. Hence, the second answer must be a lie.
Either way, after having obtained the answer to the first question, we will know for sure whether the second answer is going to be the truth or a lie.