Since the 1960s Mastermind has been studied for the combinatorial and information theoretical interest the game has to offer. Many results have been discovered starting with Erd\H{o}s and R\'enyi determining the optimal number of queries needed for two colors. For $k$ colors and $n$ positions, Chv\'atal found asymptotically optimal bounds when $k \le n^{1-\epsilon}$. Following a sequence of gradual improvements for $k \geq n$ colors, the central open question is to resolve the gap between $\Omega(n)$ and $\mathcal{O}(n\log \log n)$ for $k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$.

翻译：自20世纪60年代以来，猜数字大师（Mastermind）因其提供的组合与信息理论趣味而受到研究。自厄多斯（Erdős）和雷尼（Rényi）确定两种颜色的最优查询次数起，诸多成果被陆续发现。针对k种颜色和n个位置，赫瓦塔尔（Chvátal）在k ≤ n^{1-ε}时找到了渐近最优边界。随着对k ≥ n种颜色的逐步改进，核心未解问题在于解决k=n时Ω(n)与O(n log log n)之间的差距。本文通过提出首个在k=n情况下使用线性查询次数的算法，填补了这一差距。由此，我们得以确定任意参数k和n下猜数字大师游戏的查询复杂度。