Deadly Doors
You are locked in a room that has three doors. Two doors have bombs that trigger when the doorknob turns. One door leads out of the room. One man stands at each door. One of the men lies about what he says, and two of the men tell the truth. Man 1 stands at door 1, man 2 stands at door 2, and man 3 stands at door 3.
Here is what each man says:
Man 1: Man 2 is a liar
Man 2: Door 2 leads out of the room
Man 3: Door 1 does not lead out of the room
Which door lead out of the room?
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If Man 1 is the liar then what Man 2 and Man 3 say is consistent with Door 2 being the way out, (with both of them telling the truth).
If Man 2 is the liar then what Man 1 and Man 3 say is consistent with Door 3 being the way out, (with both of them telling the truth). This is because if Man 2 is lying then Door 2 is not the way out, and if Man 3 is telling the truth then Door 1 is not the way out either, leaving Door 3 as the only viable option out.
So we have two possible scenarios that are consistent with the given conditions, so there is no way to be certain of which door is safe.
(Note: If Man 3 is a liar then at least one of Men 1 or 2 must also be lying, in which case we would not have 2 men telling the truth as required. Thus we can narrow the choice down to Door 2 or Door 3, which gives us a 50% chance of surviving.)