Which of the following statements are true?
1. The third statement is true.
2. The above statement is false.
3. Exactly one of the above 2 statements are false.
4. Exactly 6 statements in this list are false.
5. Exactly 7 statements in this list are false.
6. Exactly 8 statements in this list are false.
7. Exactly 9 statements in this list are false.
8. The fourth statement is true.
9. The fifth statement is true.


How many and which of the statements are true?
For any clarification, comment down below.

  • 1
    $\begingroup$ Where you say "is this list" in statements 4-7, do you mean "in this list"? $\endgroup$
    – hexomino
    Commented Sep 7, 2020 at 7:59
  • 1
    $\begingroup$ @hexomino yes. that's a typo. sorry for that $\endgroup$ Commented Sep 7, 2020 at 8:06

1 Answer 1

  • Statements 1 and 2 contradict each other, so one of them is true and the other one is false.

  • That means statement 3 is true, therefore statement 1 is true and statement 2 is false.

  • We have two true statements already, so statement 6 is false and statement 7 is false.

  • Statements 4 and 8 are equivalent, so either both true or both false. Ditto for statements 5 and 9.

  • Therefore, exactly two of the first three statements are true, and either zero or two of the remaining statements are true. That makes overall either two or four true statements, so either five or seven false ones.

  • Five false statements means all the last six statements must be false, contradiction. Seven false statements means statements 5 and 9 are true, contradiction.

So indeed it is

a paradox.


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