Five statements counting their own falsehoods

Question

A card has five statements printed on it, each numbered:

Exactly one of these five statements is false.
Exactly two of these five statements are false.
Exactly three of these five statements are false.
Exactly four of these five statements are false.
All five of these statements are false.

Which of the five statements are true, and which are false?

Accepted Answer

Each statement names a different number of false statements (one, two, three, four, or five). But there is only one true answer to "how many of these are actually false." So at most one statement can be true, the one that happens to match the real count. The other four must be false. Can zero of them be true (all five false)? If all five were false, then the real number of false statements would be five, which is precisely what statement 5 claims. That would make statement 5 true, contradicting "all false." So it is impossible for all of them to be false. So exactly one statement is true and …

Five statements counting their own falsehoods

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