Which of the following statements is true?
1. Exactly one of these ten statements is false.
2. Exactly two of these ten statements are false.
3. Exactly three of these ten statements are false.
4. Exactly four of these ten statements are false.
5. Exactly five of these ten statements are false.
6. Exactly six of these ten statements are false.
7. Exactly seven of these ten statements are false.
8. Exactly eight of these ten statements are false.
9. Exactly nine of these ten statements are false.
10. Exactly ten of these ten statements are false.
Which of the above statements will be true and which of them will be false, if you remove the word 'Exactly' from all the statements ?
Ravi, a lovely problem of self referential statements! A paradox? Not necessarily, although self reference is infamous for leading to paradoxes. Stop me before I rant on about Goedel's theorem...
The first part ('exactly'): we have ten mutually exclusive statements, so only one (or none) can be true, leaving us with #9 or #10. If #10 is true, then #10 is false, no good. So this leaves us with #9.
The second part: no easy wait out this time. Assuming #1-#5 are true and the rest false 'works', that is, it is consistent. But I suppose one has to prove that this is the only answer, as the truth of any statement is only defined by the assumptions about the truths of the rest of the statements...
self reference
