A query game is a pair of a set $Q$ of queries and a set $\mathcal{F}$ of functions, or codewords $f:Q\rightarrow \mathbb{Z}.$ We think of this as a two-player game. One player, Codemaker, picks a hidden codeword $f\in \mathcal{F}$. The other player, Codebreaker, then tries to determine $f$ by asking a sequence of queries $q\in Q$, after each of which Codemaker must respond with the value $f(q)$. The goal of Codebreaker is to uniquely determine $f$ using as few queries as possible. Two classical examples of such games are coin-weighing with a spring scale, and Mastermind, which are of interest both as recreational games and for their connection to information theory. In this paper, we will present a general framework for finding short solutions to query games. As applications, we give new self-contained proofs of the query complexity of variations of the coin-weighing problems, and prove new results that the deterministic query complexity of Mastermind with $n$ positions and $k$ colors is $\Theta(n \log k/ \log n + k)$ if only black-peg information is provided, and $\Theta(n \log k / \log n + k/n)$ if both black- and white-peg information is provided. In the deterministic setting, these are the first up to constant factor optimal solutions to Mastermind known for any $k\geq n^{1-o(1)}$.

翻译：查询游戏由一组查询$Q$和一组函数（或码字）$f:Q\rightarrow \mathbb{Z}$构成。我们将此视为双人游戏：一方为“编码者”，选定隐藏码字$f\in \mathcal{F}$；另一方为“破译员”，通过依次提问$q\in Q$来推断$f$，每次提问后编码者必须回答$f(q)$的值。破译员的目标是使用尽可能少的查询次数唯一确定$f$。此类游戏的经典例子包括弹簧秤称硬币和“大师思维”（Mastermind），它们既作为娱乐游戏受到关注，也与信息论密切相关。本文提出一个求解查询游戏短程解的一般性框架。作为应用，我们给出了称硬币问题变种查询复杂度的新自包含证明，并证明了以下新结果：当仅提供黑色插针信息时，具有$n$个位置和$k$种颜色的“大师思维”确定性查询复杂度为$\Theta(n \log k/ \log n + k)$；当同时提供黑色和白色插针信息时，复杂度为$\Theta(n \log k / \log n + k/n)$。在确定性设定下，这些是针对任意$k\geq n^{1-o(1)}$的“大师思维”问题首次达到常数因子最优解的结果。