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## Homework Statement

We have 100 statements, where the nth statement says that "exactly n of these statements are false." What conclusions can you draw about the truth values of the statements?

## Homework Equations

## The Attempt at a Solution

Immediately we see that the 100th statements can't be true, because that leads to a contradiction, since all statements would be false. Statements 1-98 are false because they all imply that there would be more than one statement true. However, only one of the statements can be true at one time (because of the word "exactly"). We have shown that statements 1-98 and 100 are false by contradiction, so that leaves us with statement 99. I am confused about how we "prove" that this statement is true. For all of the others, we assumed that they were true, and this led to a contradiction, so they became false. However, for the 99th statement, if we let it be false, it doesn't seem like a contradiction occurs where it would have to be true. Thus it seems that it can be true or false. How should I be thinking about this in order to give a definite answer to the problem?