Magic: Shuffle face down cards after opponent looked at one?
No, you can't do that.

From Comprehensive Rules (bold mine):


  707.6. If you control multiple face-down spells or face-down permanents, you must ensure at all times that your face-down spells and permanents can be easily differentiated from each other. This includes, but is not limited to, knowing the order spells were cast, the order that face-down permanents entered the battlefield, which creature attacked last turn, and any other differences between face-down spells or permanents. Common methods for distinguishing between face-down objects include using counters or dice to mark the different objects, or clearly placing those objects in order on the table.