Problem:
Five friends make, respectively, the following statements.
"However one of us is chosen, the other 4 are lying".
"However one of us is chosen, the other 4 tell the truth".
"However one of us is chosen, there is another one who tells the truth".
"There is one of us such that every other one tells the truth".
"There is one of us such that every other one is lying".
Which of the following statements can be deduced from the previous ones?
Pick one
Solution:
The answer is (A). The first statement cannot be true, because by choosing one of the others it contradicts itself.
Therefore the second one cannot be true either, because by choosing the one who says it, there is the first one who is lying. Therefore the fourth one cannot be true either, because there are already two false ones.
Finally the third and the fifth contradict each other directly, so one and only one of the two is true (in particular the fifth one is true).