In this work we determine the metric dimension of ${\mathbb{Z}}_n\times {\mathbb{Z}}_n\times {\mathbb{Z}}_n$ as $\lfloor 3n/2\rfloor$ for all $n\ge 2$. We prove this result by investigating a variant of Mastermind.

Mastermind is a famous two-player game that has attracted much attention in the literature in recent years. In particular we consider the static (also called non-adaptive) black-peg variant of Mastermind. The game is played by a codemaker and a codebreaker. Given c colors and p pegs, the principal rule is that the codemaker has to choose a secret by assigning colors to the pegs, i.e., the secret is a p-tuple of colors, and the codebreaker asks a number of questions all at once. Like the secret, a question is a p-tuple of colors chosen from the c available colors. The codemaker then answers all of those questions by telling the codebreaker how many pegs in each question are correctly colored. The goal is to find the minimal number of questions that allows the codebreaker to determine the secret from the received answers. We present such a strategy for this game for $p=3$ pegs and an arbitrary number $c\ge 2$ of colors using $\lfloor 3c/2\rfloor +1$ questions, which we prove to be both feasible and optimal.

The minimal number of questions required for p pegs and c colors is easily seen to be equal to the metric dimension of ${{\mathbb{Z}}_c}^p$ plus 1 which proves our main result.

在这项工作中，我们确定 ${\mathbb{ž}}_{ñ}\times {\mathbb{ž}}_{ñ}\times {\mathbb{ž}}_{ñ}$ 如 $\lfloor 3ñ/2\rfloor$ 对全部 $ñ\ge 2$。我们通过研究Mastermind的变体来证明这一结果。

Mastermind是著名的两人游戏，近年来受到文学界的关注。特别是，我们考虑了Mastermind的静态（也称为非自适应）黑钉变体。该游戏由一个代码制作者和一个代码破坏者进行。给定c种颜色和p钉，主要规则是代码制作者必须通过将颜色分配给peg来选择秘密，即，秘密是p色的元组，并且代码破解者立即提出许多问题。像秘密一样，问题是从c中选择的p个元组可用的颜色。然后，代码制作者通过告诉代码破解者每个问题中多少个钉子正确着色来回答所有这些问题。目的是找到最少数量的问题，以使代码破解者能够从收到的答案中确定秘密。我们为此游戏提出了这样的策略$p=3$ 钉子和任意数量 $C\ge 2$ 使用的颜色 $\lfloor 3C/2\rfloor +\mathrm{1个}$ 问题，我们证明这既可行又最优。

的所需的问题最小数量p栓和Ç颜色是很容易看到为等于的度量尺寸${{\mathbb{ž}}_C}^p$ 加1证明了我们的主要结果。