Under a certain legal system, prisoners are allowed to make one statement. If their statement turns out to be true then they are hanged. If the statement turns out to be false then they are shot. One prisoner made a statement and the judge had no option but to set him free. Which one of the following could be that statement?

To determine which statement forces the judge to set the prisoner free, we must analyze the rules of the legal system and how they apply to each possible statement. The rules dictate the following outcomes based on the truth value of the prisoner's statement:
1. If the statement is true, the prisoner must be hanged.
2. If the statement is false, the prisoner must be shot.

Let us evaluate the consequences of the statement: "I will be shot".

Case 1: Assume the statement is true.
If the statement "I will be shot" is true, then by the system's rules, a true statement requires the prisoner to be hanged. However, if the prisoner is hanged, the statement "I will be shot" becomes false. This creates a contradiction, so the statement cannot be true if the outcome is hanging.

Case 2: Assume the statement is false.
If the statement "I will be shot" is false, then by the system's rules, a false statement requires the prisoner to be shot. However, if the prisoner is shot, the statement "I will be shot" becomes true. This also creates a contradiction, so the statement cannot be false if the outcome is shooting.

Because both assumptions (that the statement is true and that the statement is false) lead to logical contradictions (paradoxes), the judge cannot execute the prisoner by hanging or by shooting. To maintain the logical consistency of the legal rules, the judge has no choice but to set the prisoner free.