A sheet of paper has statements numbered from 1 to 100. For each $N$, $1\leq N \leq100$, Statement $N$ says, "Exactly $N$ of the statements in this sheet are false". Which statements are true?
I can't seem to figure out the answer as every statement seems to contradict rest of the others. Can someone help me on this?
Best Answer
You are correct in that each of the statements contradict one another. So at most one of the statements is true.
If none of the statements are true, then all $100$ of them are false. However that would make the $100$th statement true, which is a contradiction.
Hence the only possibility is that exactly one statement is true and $99$ are false. If this is the case, the the $99$th statement is true and the remaining $99$ are indeed false.