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19 Posts

Posted - 02/01/2012 :  13:18:46  Show Profile  Reply with Quote
Consider the following ten statements. 1. Exactly one of these statements is false. 2. Exactly two of these statements are false. 3. Exactly three of these statements are false. 4. Exactly four of these statements are false. 5. Exactly five of these statements are false. 6. Exactly six of these statements are false. 7. Exactly seven of these statements are false. 8. Exactly eight of these statements are false. 9. Exactly nine of these statements are false. 10. Exactly ten of these statements are false. How many of the statements are true?
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Advanced Member

1469 Posts

Posted - 02/01/2012 :  16:42:09  Show Profile  Reply with Quote
Take statements 1 and 2. They both can't be true at the same time because you can't have EXACTLY one and EXACTLY two at the same time. If ONE statement is false, then that says it, ONE statement is false (it isn't two). It is the same situation for any combination of the statements. Only one could be true at a time. That would eliminate #1 itself from being true because #1 being true would state that 9 of the statements are true, which can't happen because of what I already stated above. It also eliminates #2 from being true because that would say 8 of the statements are true.
And so on up to statement #8 (that would say two of them are true and that is impossible).
So, if one of them is true, which one? Look at #9. If it is true, that makes all the all the others false (because there are only 9 other statements and it says 9 statements are false). Does that cause a problem? No. Statement 10 would be false because NOT exactly ten would be false (9 of the would be).
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