probability logic

Question

A notebook contains only hundred statements as under:

1 . This notebook contains 1 false statement. 2 . This notebook contains 2 false statements . . .
. 99 . This notebook contains 99 false statements.
100. This notebook contains 100 false statements.

Which of the statements is correct? 1. 100th
2. 1st
3. 99th
4. 2nd

Answer 1

If the $n^{th}$ statement is correct, that means that there must be $n$ false statements. However, note that two of the statements cannot both be true at the same time, as otherwise for $m \neq n$ this would imply that $m = n$, a contradiction. So, we must have at most one true statement. Also, note that if we had no true statements, statement 100 would be true, a contradiction. So there is exactly one true statement, so that $n + 1 = 100$ and $n = 99$, meaning that the $99^{th}$ statement is true.

Answer 2

Yushwuth's answer is correct in the sense that it gives the only consistent assignment of truth values to the statements. However, self-referential statements such as these are tricky. My notebook contains the following $100$ statements.

1) $1$ of these statements is false and you owe me \$100.

2) $2$ of these statements are false and you owe me \$100.

...

99) $99$ of these statements are false and you owe me \$100.

100) $100$ of these statements are false and you don't owe me \$100.

If you don't owe me \$100, statements (1) to (99) are all false, but then (100) leads to a contradiction whether it is true or false. So that can't be right.

If you do owe me \$100, statement (100) is false, and at most one of the others is true. It could be that (99) is true or that none is true; I don't care which, but I would like that \$100 please...

Answer 3

At first we will show that more than one statement cannot be correct simultaneously. This is easy to see since any two persons among the 100 contradict each other in their statements. Now there arises two possibilities: (a) All statements are false (b) exactly 1 statement is true We can reject (a) as the 100th person says so (so 100th person must be correct). Hence only one statement is correct. Let the nth statement be correct. nth statement says exactly n statements are incorrect and we know 1 statement is correct. So n + 1 = 100. Hence n = 99. Thus the 99th is true.