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 $k\ge n$ colors and $\ell :=k+1-n$ , we prove a lower bound of ${\sum }_{j=\ell }^k{log}_{2}j$ and an upper bound of $n{log}_{2}n+k$ 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 $\mathsf{\Theta }(nlogn)$ queries.