1. Sentence 2 is true.
2. An odd number of prime-numbered sentences are true.
3. A prime number of these sentences are true.
4. Sentence 1 is true.
5. A prime number of these sentences are false.
Which sentences are true and which sentences are false?
Sentences 3 and 5 are true. Sentences 1,2, and 4 are false.
EXPLANATION:
Case 1: No true sentences, all 5 false sentences.
In this case, Sentence-5 is true, being a prime number of false sentences. Contradiction.
Case 2: 1 true sentence, 4 false sentences
In this case, sentences-2 & 3 are false, so that sentences 1 & 4 are also false. Also, 4 false sentences make sentence 5 false. So, all statements are false. Contradiction.
Case 3: 2 true sentences, 3 false sentences
2 true sentences make sentence 3 true but sentence 2 false. This renders sentences 1 and 4 as false. Again, three false sentences renders sentence 5 as true.
Thus, there are 2 true sentences and 3 false sentences.
Case 4: 3 true sentences, 2 false sentences
3 true sentences imply that sentence 2 and sentence 3 are true. Then, sentences 1 & 4 are also true. So there are at least 4 true sentences. This is a contradiction.
Case 5: 4 true sentences, 1 false sentence.
4 true sentences give sentences 1 to 4 as false.
So, there are more than 1 false sentence. Contradiction.
Case 6: All 5 true sentences, no false sentence.
No false statement makes sentence 5 as false. This is a contradiction.
Now, considering Cases 1 to 6 we observe that only Case 3 is valid.
Consequently, sentences 3 & 5 are true, and sentences 1,2, and 4 are false.
Edited on February 9, 2022, 7:34 am
Edited on February 9, 2022, 7:40 am
Puzzle Solution
