Puzzle : Truant List of Statements I was working my way through some puzzles in Discrete Maths by Rosen, when I came across the following question:

The $n^{th}$ statement in a list of 100 statements is : "Exactly $n$ of the statements in this list are false"
  
  
*
  
*What conclusion can you draw from these statements ?
  
*Answer the first part if the $n^{th}$ statement is : "At least $n$ of the statements in this list are false" ?
  
*Answer the second part assuming that the list contains 99 statements ?

My Solution (Inadequate):

  
*
  
*The 99th Statement is True and the rest are false
  
*I am all thumbs for the next two parts

Book solution:

  
*
  
*The 99th Statement is True and the rest are false
  
*Statements 1 through 50 are all true and statements 51 through 100 are all false
  
*This cannot happen; it is a paradox, showing that these cannot be statements.
  

My question:
Why is this so?
 A: For the first part, "Exactly $n$ of the statements are false", note that they all contradict each other, so at most one is true, so at least $99$ are false.  This guarantees that $1-98$ are false.  You can have $99$ being true and $100$ false and all is consistent.  If $99$ if false as well we cannot assign a consistent truth value to $100$, so the only assignment is number $99$ true and all others false.   
For $100$ sentences saying "At least $n$ of these are false" we cannot have $1$ be false because its negation is all of these are true, so it would have to be true, in conflict with our assumption that it is false, so $1$ is true and at least one of the statements is false.  Now we know that statement $100$ is false because one of the statements is true.  Statement $2$ cannot be false, because its falsity would require at most one true statement, which is $1$, which would make it true.  We continue back and forth, showing the high statement numbers are false and the small ones true until we show $1-50$ are true and $51-100$ are false.  
If there were $99$ sentences of the form "At least $n$ of these are false" we proceed just like the $100$ case until we have $1-49$ true and $51-99$ false.  Then if $50$ is true there are only $49$ false statements and if it is false there are $50$.  Either way is a contradiction and we conclude there is no consistent truth assignment.
A: Answer the first part if the nth statement is : "At least n of the statements in this list are false"
Suppose the $1st$ statement is false, then it is not the case that at least $1$ of the stamenets are false. This is a contradiction as the $1st$ statement is false.
So, the first statement must be true. You can repeat this argument all the way up to (and including) 50, but what happens at 51?
If the $51st$ statement is true, there must be at least $51$ false statements. As 1-50 are true, this cannot be the case and therefore statement $51$ must be false. The same argument applies to 51-100.
Answer the second part assuming that the list contains 99 statements
You can use the same argument as before for proving that $1-49$ are true.
What about statement $50?$?


*

*If it is true, there must be at least $50$ false statements. As $1-50$ are true in this case, this can never be the case as we only have $99$ statements

*If it is false, it is not the case that at least $50$ of the statements are false. As $1-49$ are true, this can only be the case if at least one of the statements in the interval $52-99$ is true. This can never be the case for the above reason. Thus statement $51$ is neither true or false.

