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 ?
Okay, while it is generally the case that a proff involves some rigorous procedure to show that some way is the only way, is it not just as valid to show all the possibilities that are false, and then show there is only one possibility left, and THEN show that this possibility is true? OK, that seems even to me a bit fuzzy, but I'll expand on this scenario in reference to this problem, by giving what I think is a valid (method for) proof of the "at least" scenario...
1. Note that a statements value of true forces all lower numbered statement to have a value of true.
2. Note the reverse as well - a false statement forces all higher number statements to be false.
3. This now leaves us with only 11 possibilities for statement true/false-ness, of the original 2^10 possibilities, much more manageable by trial and error.
4. Evaluate the situation for each of these cases, which gives a unique possibility for satisfying the requirements of the problem.
Does this not count as a proof - despite lacking much structure, the end result is that a single unique solution has been found, whil all others have been ruled out. Or are the theorum police (postulate police, lemma police etc.) on their way to arrest me for proof purgery?