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In this paper we determine the metric dimension of \( K_a \times K_b \times K_c\) for all \(a,b,c\in \mathbb {N}\) with \( a \le b \le c \) as follows. For \(3a&lt;b_c_ and="" _2b="" _le="" c_="" _="" this="" value="" is="" _c-1_="" for="" _3a_b_c_=""&gt; c\) , it is \(\left\lfloor \frac{2}{3}(b+c-1) \right\rfloor \) , and for \(3a=b+c\) , it is \(\left\lfloor \frac{a+b+c}{2} \right\rfloor -1 \) . The only open case is \(3a&gt;b+c\) , where two values are possible, namely \(\left\lfloor \frac{a+b+c}{2} \right\rfloor -1 \) and \(\left\lfloor \frac{a+b+c}{2} \right\rfloor \) . This result extends previous results of&#xa0;[4], who computed the metric dimension of \( K_a \times K_b\) , and of&#xa0;[14], who computed the metric dimension of \( K_a \times K_a \times {K_a}\) . We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows,&#xa0;as the number of questions of these strategies&#xa0;is equal to the metric dimension of \(K_a \times K_b \times K_c\) .</b_c_>

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Determining the Metric Dimension of  \( K_a \times K_b \times K_c \) by Static Black-Peg Mastermind

  • 会议论文

  • 发表日期：

    2026年

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  • Valentin Gledel,
  • Gerold Jäger

摘要

In this paper we determine the metric dimension of \( K_a \times K_b \times K_c\) for all \(a,b,c\in \mathbb {N}\) with \( a \le b \le c \) as follows. For \(3a<b_c_ and="" _2b="" _le="" c_="" _="" this="" value="" is="" _c-1_="" for="" _3a_b_c_=""> c\) , it is \(\left\lfloor \frac{2}{3}(b+c-1) \right\rfloor \) , and for \(3a=b+c\) , it is \(\left\lfloor \frac{a+b+c}{2} \right\rfloor -1 \) . The only open case is \(3a>b+c\) , where two values are possible, namely \(\left\lfloor \frac{a+b+c}{2} \right\rfloor -1 \) and \(\left\lfloor \frac{a+b+c}{2} \right\rfloor \) . This result extends previous results of [4], who computed the metric dimension of \( K_a \times K_b\) , and of [14], who computed the metric dimension of \( K_a \times K_a \times {K_a}\) . We prove our result by introducing and analyzing a new variant of Static Black-Peg Mastermind, in which each peg has its own permitted set of colors. For all cases, we present strategies which we prove to be both feasible and optimal. Our main result follows, as the number of questions of these strategies is equal to the metric dimension of \(K_a \times K_b \times K_c\) .