permutation search game

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Arrange the natural numbers $1$ through $n$ in a random order (the order is unknown and has a uniform distribution). Now make a sequence of guesses as to which number is in which slot, one number and one slot at a time. You will be told after each guess whether it is correct or not. The game ends when the order of the $n$ numbers has been determined. For the worst case and the average case, respectively, is there a strategy that takes fewer guesses than the trivial elimination?