Playing Detective

Question

Four suspects were assembled in the director's office, having been accused of a devious crime: turning off the light switch during Mr. Buehler's business presentation. It was known that only one of the four turned off the switch. All four were friends, and the director's secretary overheard them plotting before they were brought into the director's office. They all agreed to tell the same number of false statements, although the secretary did not hear the agreed-upon number. Their statements are below. Who turned off the light switch?

Joe: -Frank didn't do it. -I went to college with Felipe -I didn't do it

Felipe: -I didnt do it -Joe didnt go to college with me -John didnt do it

John: -We all agreed to tell one false statement -i didnt do it -Felipe did it

Frank: -We all agreed to tell two false statements. -Felipe didnt do it - i didnt do it

Answer 1

It can't all be true because there are contradictions and it can't all be false because, by looking at Joe, Frank would have done it and Joe would have done it.

Therefore the options are 1 or 2 false statements.

Looking at John, either way "We all agreed to tell one false statement" goes, either "[John] didn't do it" or "Felipe did it" is false, and the other is then true. From this, we learn "Felipe did it" is false and "[John] didn't do it" is true, so either Joe or Frank did it. By looking at Felipe, we then learn that there is just one false statement.

Then, by looking at Frank we see that Joe did it. (and checking that there are still no contradictions by looking at Joe)

Answer 2

Frank's first statement can't be true, because then his second and his third would be both false, which is impossible. Since the number of false statement can't be either $0$ or $3$, it must be $1$. Then Frank's second and third statements are true, and Felipe and Frank are ruled out. John's last statement is false, so his second is true, and he is ruled out. The only one left is Joe.