How Many Truth-tellers?

Relevant wiki: Truth-Tellers and Liars

If there are 0 truth-tellers, Alex would be a truth-teller, stating "0 or 1." However, that would mean there is 1 truth-teller which contradicts the original assumption "There are 0 truth-tellers."

If there is 1 truth-teller, Alex and Bella would be truth-tellers, stating "0 or 1" and "1 or 2". However, that would mean there are 2 truth-tellers which contradicts the original assumption "There is 1 truth-teller."

If there are 2 truth-tellers, Bella and Chris would be truth-tellers, stating "1 or 2" and "2 or 3". This means Bella and Chris are the 2 truth-tellers and the original assumption is satisfied.

If there are 3 truth-tellers, Chris and David would be truth-tellers, stating "2 or 3" and "3 or 4". However, that would mean there are 2 truth-tellers which contradicts the original assumption "There are 3 truth-tellers."

If there are 4 truth-tellers, David would be a truth-teller, stating "3 or 4.". However, that would mean there is 1 truth-teller which contradicts the original assumption "There are 4 truth-tellers."

Therefore, the only possible number of truth-tellers is $\boxed{2}$ .

I thought about the problem more generally:

Suppose N ≥ 3 people respond to the question, "How many among you are truth-tellers?", and akin to the scenario in Tarmo's original problem, suppose:

• the first person says, "0 or 1."
• the second says, "1 or 2."
• and so on, until the last person says, "N-1 or N."

Now, the number of truth-tellers is at least 0 and at most N. Let's embark on a case analysis:

Case 1: If the number of truth-tellers is 0 or N, then 1 statement is true —namely the first or last— and the rest are false.

Case 2: If the number of truth-tellers is strictly between 0 and N, then 2 consecutive statements are true and the rest are false. (For example, if there are 100 people and 5 are truth-tellers, the only true statements are "4 or 5" and "5 or 6".)

Thus by case analysis, we conclude that the number of truth-tellers is 1 or 2 — which is precisely the second person's answer. ${}^\dagger$ Therefore the second person is a truth-teller.

The above case analysis also tells us that if a statement other than the first or last is true, then we must be in the second case in which 2 statements are true. This is indeed the case we are in, for the second statement is true. ${}^\ddagger$ Hence exactly $\boxed{2}$ statements are true, namely "1 or 2" and "2 or 3".

$\dagger$ If N = 1, there is no second person, so the argument fails at this point. In this case, there is only one person, and that person says, "0 or 1.", which is true.

$\ddagger$ If N = 2, the second person is the last person, so the argument fails at this point. Indeed, in this case, the problem statement is inconsistent.

Consider the generalization described by Jeremy Weissmann, with $N$ persons.

Since there are $N$ persons, the count of truth tellers is a number in the range 0 through $N$ . Given all the answers, there must be one or two persons who told the truth, since each count is mentioned either once (viz. 0 and $N$ ) or twice.

If the count would be one, then there would be two truth-tellers (the two persons who said the disjunct “1”). Contradiction! So, the count equals 2, and the truth tellers are the ones answering “1 or 2” and “2 or 3”.

The only quantity that matches with its own frequency (of how many times it was mentioned in the interview conversation) is 2, so Bella and Chris are 2 truth tellers while Alex and David are 2 liars.

• OR statement would be true if there exist at least one truth within it.