General conjecture on the optimal covering trails for any $k$-dimensional cubic lattice
Résumé
We introduce a general conjecture involving minimum-link covering trails for any given $k$-dimensional grid $n \times n \times \cdots \times n$, belonging to the cubic lattice $\mathbb {N}^{k}$. In detail, if $n$ is above two, we hypothesize that the minimal link length of any covering trail, for the above-mentioned set of $n^k$ points in the Euclidean space $\mathbb{R}^k$, is equal to $h(n,k)=\frac{n^k-1}{n-1}+c \cdot (n-3)$, where $c = k-1$ iff $h(4,3)=23$, $c = 1$ iff $h(4,3)=22$, or even $c = 0$ iff $h(4,3)=21$.
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