Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper Here's a problem presented in my proofs class that I cannot for the life of me figure out. Apparently the answer is P2 according to the answer book but I have no idea how Help is much appreciated, thanks in advance!
Anna wrote down the following 3 statements, denoted P1, P2, P3, on a blank piece of paper:
P1 : There is exactly one FALSE statement written on this piece of paper.
P2 : There are exactly two FALSE statements written on this piece of paper .
P3 : There are exactly three FALSE statements written on this piece of paper.
One of the above statements is TRUE. What statement is TRUE?
 A: Hint. If exactly one of the above statements is TRUE, and there are exactly three statements in all, then exactly how many are FALSE?
A: Suppose that P1 is True. Then P2 and P3 must be false, a contradiction to P1.  
If you follow the same reasoning for each statement, you will find your answer.
A: Try case by case and see the only one who can happen.
If P1 is true than exist just one false, so we have two options P2-T and P3-F; or P2-F and P3-T; note that both are impossible. Then we have P1 must be false (because we showed that in every case of assuming P1 true, contradictions happens).
Now we have P1-F. And assume that P2 is true, so must exist exactly to falses, how we already have P1-F this simply leads us to P3-F (P1-F P2-T P3-F), this configuration is possible! And we'll see that its the only one.
Still assuming P1-F, suppose we had P2-F too, then or P3-T or P3-F, note that both cases lead us to a contradiction. (P1-F P2-F P3-T but there are two falses, not three) and (P1-F P2-F P3-F, there are EXACTLY three falses) so both can't happen.
Now note that the cases are over and the only possible is (P1-F P2-T P3-F)
A: You're told that one of these statement — which means precisely one of them — is TRUE. So you can simply consider all possible cases in a very straightforward manner:


*

*maybe P1 is TRUE and the other two are FALSE;

*or maybe P2 is TRUE and the other two are FALSE;

*or maybe P3 is TRUE and the other two are FALSE.


When you consider these three possible cases, you'll see that only one of them works, while the other two lead to a contradiction.
EDIT. Note that we don't even need to be told that one of them is true. Even without this additional information there's only one non-contradictory way (the same way) to assign TRUE/FALSE values to these three statements.
