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I think this is lost in translation. Do you know who the liar is ?
I am going to go out on a whim and state that not one person on this forum believes in time travel.
Is there not a "Zen" diagram or some similar named construction with intersecting circles for solving such logic problems.
... but I like the idea of a Zen diagram,
"If you were the other person, which road would you tell me leads to Rome?" They would both tell me the wrong road, so I'd then use the other one.