The exact query complexity of yes-no permutation mastermind

Mastermind is famous two-player game. The first player (codemaker) chooses a secret code which the second player (codebreaker) is supposed to crack within a minimum number of code guesses (queries). Therefore, the codemaker's duty is to help the codebreaker by providing a well-defined error measure between the secret code and the guessed code after each query. We consider a variant, called Yes-No AB-Mastermind, where both secret code and queries must be repetition-free and the provided information by the codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and k≥n colors and ℓ:=k+1−n , we prove a lower bound of ∑kj=ℓlog2j and an upper bound of nlog2n+k on the number of queries necessary to break the secret code. For the important case k=n , where both secret code and queries represent permutations, our results imply an exact asymptotic complexity of Θ(nlogn) queries.