The Answer Is Right There

Relevant wiki: Truth-Tellers and Liars

There is only $\boxed{1}$ statement, statement E , which is true as explained below:

• Statement A: "All of those below" is false, because statements B through F cannot be all true.
• Statement B: "None of those below" is false, because statement E is true .
• Statement C: "All of those above" is false, because statements A and B are false.
• Statement D: "Exactly one of the above" is false, because all statements A, B and C are false.
• Statement E: "None of the above" is true , because all statements A, B, C and D are false.
• Statement F: "None of the above" is false, because statement E is true .

Here's a pathway to finding this answer:

If A is true, then F is true, but F says that A is false. Similarly, if F is true, then A is false, but A says that F is true. So, because of these contradictions, both A and F are false.

C is therefore immediately false (since A is false).

If B is true, then D is true. But if D is true, then B is false. This is a contradiction, so B is false, and it follows that D is false too.

This leaves E. If E is true, this checks out: exactly 1 statement is true, and it is E. E cannot be false, because then F would be true.