有和无白钉答案的广义AB对策的最坏情况问题数
The Worst Case Number of Questions in Generalized AB Game with and without White-peg Answers
摘要：
AB游戏是一个两人游戏，密码制作者必须选择一个密码，而密码破译者必须在尽可能少的问题中猜测它。它是著名的主谋游戏的变体，唯一的区别是，秘密和问题中的所有挂钩都必须具有不同的颜色。在这项工作中，我们考虑了广义AB游戏，其中对于给定的任意数$p$，$c$与$p \le c$的密码由$p$钉每个$c$颜色之一，答案只包括一些黑色和白色钉。在那里，黑色挂钩的数量等于在相应问题中匹配的挂钩的数量以及在位置和颜色上的秘密，并且白色挂钩的数量等于在相应问题中匹配的挂钩的额外数量以及仅在颜色上的秘密。我们还考虑了广义AB博弈的一个变体，其中省略了白钉的信息。这个变体被称为广义Black-peg AB博弈。设$\ab(p,c)$和$\abb(p,c)$分别为广义AB对策和广义Black-peg AB对策的最坏情况问题数。结合计算机程序的理论考虑，我们确认已知的精确值的$\ab(2,c)$和$\ab(3,c)$和证明紧界$\ab(4,c)$。此外，我们提出了精确的值为$\abb(2,c)$和$\abb(3,c)$和紧界为$\abb(4,c)$。
英文摘要：
The AB game is a two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. It is a variant of the famous Mastermind game, with the only difference that all pegs in both, the secret and the questions must have distinct colors. In this work, we consider the Generalized AB game, where for given arbitrary numbers $p$, $c$ with $p \le c$ the secret code consists of $p$ pegs each having one of $c$ colors and the answer consists only of a number of black and white pegs. There the number of black pegs equals the number of pegs matching in the corresponding question and the secret in position and color, and the number of white pegs equals the additional number of pegs matching in the corresponding question and the secret only in color. We consider also a variant of the Generalized AB game, where the information of white pegs is omitted. This variant is called Generalized Black-peg AB game. Let $\ab(p,c)$ and $\abb(p,c)$ be the worst case number of questions for Generalized AB game and Generalized Black-peg AB game, respectively. Combining a computer program with theoretical considerations, we confirm known exact values of $\ab(2,c)$ and $\ab(3,c)$ and prove tight bounds for $\ab(4,c)$. Furthermore, we present exact values for $\abb(2,c)$ and $\abb(3,c)$ and tight bounds for $\abb(4,c)$.
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