on a sheet of paper, you have 100 statements written down. the first says, "at most 0 of these 100 statements are true." the second says, "at most 1 of these 100 statements are true." ... the nth says, "at most (n-1) of these 100 statements are true. ... the 100th says, "at most 99 of these statements are true." how many of the statements are true? is it 99 statements?
These are self-referencing statements which creates a conceptual problem for me, but the statement "at most 99 of these (preceding ) statements are true", given the "at most" qualifier, makes the 100th statement true, but vacuous. I'm thinking the first statement must be true under any circumstances since it's not preceded by any statements.
If n statements are true, then "at most k statements are true" would be false for k<n and true for k between n and 99 inclusive. Thus we must have n=99-(n-1) which yields n=50.
It would be instructive to look at the case with 4 statements. So A: at most 0 statements are true B: at most 1 statement is true C: at most 2 statements are true D: at most 3 statements are true There are a few cases to consider: 1) All the statements are false Then A would be true. So not all statements are false, which is a contradiction 2) Exactly one statement is true Then A and B would be true. This is a contradiction 3) Exactly two statements are true A and B would be false. C and D would be true. So no contradiction here. 4) Exactly three statements are true Then only D would be true. So there are no three statements true. Contradiction! 5) Exactly four statements are true Then all statements would be false. This is a contradiction. So the correct answer here is that exactly two statements are true: C and D. An analogous method would show you that, in your problem, exactly 50 statements are true (namely the last 50).