Mark with T or F all the below statements in such a way that they do not contradict with each other:
- At most $1$ statement is true
At most $1$ statement is false
At most $2$ statements are true
At most $2$ statements are false
At most $3$ statements are true
At most $3$ statements are false
At most $4$ statements are true
At most $4$ statements are false
At most $5$ statements are true
At most $5$ statements are false
We have a total of $10$ statements. If we mark the $1^{\mathrm{st}}$ with T, it means all others must be false. But this contradicts with the 2nd. We can safely mark the last $2$ as True. But if we say, for example, at most $5$ statements are true, can it be also correct that "at most $3$ (or $2$ or $4$, etc.) statements are true"? At most $5$ statements are true means at least $5$ statements are false, which is OK in combination with the $10^{\mathrm{th}}$, right?
But if we say "at most $1$ statement is true", it means "at minimum $9$ statements are false". But if "at minimum $9$ statements are false", doesn't it mean also that "at minimum $7$ statements are false" for example?
How can we combine them all together?
I am completely stuck!