Mastermind is famous two-players 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, codemaker's duty is to help 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 codemaker only indicates if a query contains any correct position at all. For this Mastermind version with n positions and $k\le n$ colors we prove a lower bound of $\log_2(k+1-n)+\log_2(k+2-n)+\dots+\log_2(k)$ 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 $\Theta(n\log_2(n))$ queries.

中文翻译：

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Yes-No Permutation Mastermind 的精确查询复杂度

Mastermind 是著名的两人游戏。第一个玩家（codemaker）选择一个密码，第二个玩家（codebreaker）应该在最少的密码猜测（查询）次数内破解。因此，codemaker 的职责是通过在每次查询后在密码和猜测代码之间提供明确定义的错误度量来帮助密码破解者。我们考虑一种称为 Yes-No AB-Mastermind 的变体，其中密码和查询都必须是无重复的，并且 codemaker 提供的信息仅指示查询是否包含任何正确的位置。对于这个有 n 个位置和 $k\le n$ 种颜色的 Mastermind 版本，我们证明了 $\log_2(k+1-n)+\log_2(k+2-n)+\dots+\log_2(k)$ 的下界以及破解密码所需的查询次数的上限 $n\log_2(n)+k$。对于重要的情况 $k=n$，