Since all the statements are (pairwise) mutually contradictory, at most one of them is an Ent. If there are no Ents, then D would be telling the truth, making him an Ent. But that is a contradiction. So there is exactly one Ent, and consequently three Bents. So C, as the only truth-teller, is an Ent, and the others are Bents.
Explanation-
To solve this problem, let's analyze the statements made by each inhabitant one by one and figure out how many of them are Bents.
Restating the inhabitants' statements:
- A: Exactly one of us is a Bent.
- B: Exactly two of us are Bents.
- C: Exactly three of us are Bents.
- D: Exactly four of us are Bents.
Ents always tell the truth, and Bents always lie. Therefore, if an inhabitant is an Ent, their statement must be true. If they are a Bent, their statement must be false.
Let's analyze each statement in the case of different numbers of Bents:
Case 1: Assume that there is 1 Bent.
- A's statement: "Exactly one of us is a Bent."
- Since there is 1 Bent, A's statement is true. A must be an Ent.
- B's statement: "Exactly two of us are Bents."
- This is false because there is only 1 Bent, so B must be a Bent.
- C's statement: "Exactly three of us are Bents."
- This is false because there is only 1 Bent, so C must be a Bent.
- D's statement: "Exactly four of us are Bents."
- This is false because there is only 1 Bent, so D must be a Bent.
In this case, A is the only Ent, and B, C, and D are all Bents. This configuration is consistent with the conditions, and it results in 3 Bents.
Case 2: Assume that there are 2 Bents.
- A's statement: "Exactly one of us is a Bent."
- This is false, so A must be a Bent.
- B's statement: "Exactly two of us are Bents."
- This is true, so B must be an Ent.
- C's statement: "Exactly three of us are Bents."
- This is false, so C must be a Bent.
- D's statement: "Exactly four of us are Bents."
- This is false, so D must be a Bent.
In this case, A, C, and D are Bents, and B is an Ent. This configuration results in 3 Bents, which is consistent with the assumption of 2 Bents. However, this doesn't work because we assumed there were 2 Bents to begin with, and the result shows 3.
Case 3: Assume that there are 3 Bents.
- A's statement: "Exactly one of us is a Bent."
- This is false, so A must be a Bent.
- B's statement: "Exactly two of us are Bents."
- This is false, so B must be a Bent.
- C's statement: "Exactly three of us are Bents."
- This is true, so C must be an Ent.
- D's statement: "Exactly four of us are Bents."
- This is false, so D must be a Bent.
In this case, A, B, and D are Bents, and C is an Ent. This configuration results in 3 Bents, which is consistent with the assumption of 3 Bents.
Case 4: Assume that there are 4 Bents.
- A's statement: "Exactly one of us is a Bent."
- This is false, so A must be a Bent.
- B's statement: "Exactly two of us are Bents."
- This is false, so B must be a Bent.
- C's statement: "Exactly three of us are Bents."
- This is false, so C must be a Bent.
- D's statement: "Exactly four of us are Bents."
- This is true, so D must be an Ent.
However, this case results in 3 Bents rather than 4. So, this case doesn't work.
Conclusion:
The only consistent solution is that there are 3 Bents (A, B, and D are Bents, while C is the Ent).
so, option (C) is correct
