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：查询游戏由查询集合$Q$与函数（或码字）$f:Q\rightarrow \mathbb{Z}$的集合$\mathcal{F}$共同构成。我们将其视为一种双人博弈：一方（编码者）选择一个隐藏码字$f\in \mathcal{F}$，另一方（解码者）通过依次提出查询$q\in Q$来试图确定$f$，每次查询后编码者必须反馈$f(q)$的值。解码者的目标是用尽可能少的查询唯一确定$f$。这类游戏的两个经典例子是弹簧秤称硬币与Mastermind，它们既作为娱乐游戏具有趣味性，也与信息论紧密相关。本文将为查询游戏中的短解寻找提供一个通用框架。作为应用，我们给出了各种变体称硬币问题查询复杂度的全新自包含证明，并证明了新结果：当仅提供黑钉信息时，具有$n$个位置和$k$种颜色的Mastermind的确定性查询复杂度为$\Theta(n \log k/ \log n + k)$；当同时提供黑钉和白钉信息时，其复杂度为$\Theta(n \log k / \log n + k/n)$。在确定性设定下，对于任意$k\geq n^{1-o(1)}$，这些结果首次给出了Mastermind的最优解（常数因子内最优）。