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You are given a sheet of paper with $100$ sentences, in the form of: There are exactly $x$ wrong sentences on this sheet.

There is exactly $1$ wrong sentence on this sheet.

There are exactly $2$ wrong sentences on this sheet.

...

There are exactly $99$ wrong sentences on this sheet.

There are exactly $100$ wrong sentences on this sheet.

Your task is find out which sentence says the truth (there can be only one).

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It depends on what you mean by the wording of the sentences: Does "There is 1 wrong sentence on this sheet" mean "There is exactly 1 wrong sentence on this sheet," or does it mean "There exists 1 wrong sentence on this sheet (but there may be more)"?

If it is the first case, the answer is 99, because there are 100 sentences total on the sheet, and all but one of them are wrong.

If it is the second case, the answer is 50. Sentences #1 to #50 inclusive are correct, and sentences #51 to #100 inclusive are incorrect.

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  • $\begingroup$ Sorry for my wording. Your first case is what i meant. I edited the answer to match the wordings. $\endgroup$
    – hetzi
    Jul 24 '14 at 8:38

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