An unbroken path of straight lines passing through $3\times 3\times 3$ lattice grid It's well known that it requires an unbroken path of 4 straight lines to cover all 9 dots in a 3 by 3 lattice grid (see here).
By making use of the 4-line path in $3\times 3$ grid, it only needs 14 lines to cover the three-dimensional $3\times 3\times 3$ grid (4 lines for each layer and 2 lines for layer connection, which yields $4\times 3 + 2 = 14$ lines).
Is it the optimal number of lines needed for this problem? What about general results for $N\times N\times N$ grid?
 A: Happy to answer this question, as the author of the solution of the $k$-dimensional $3 \times 3 \times \dots \times 3$ case.
Here is the paper with my full (constructive) proof that any optimal solution has exactly $\frac{3^k-1}{2}$ lines (i.e., a covering trail with link-length $\frac{3^k-1}{2}$: SOLVING THE 106 YEARS OLD 3^k POINTS PROBLEM WITH THE CLOCKWISE-ALGORITHM).
Thus, if $k=3$ is given, then we just need $13$ straight lines connected at their endpoints, and I proved in 2019 that this solution is optimal.

We can go even further, exploring the $n \times n \times \dots \times n$ problem, for any $n \in \mathbb{Z}^{+}$, and here are the most recent results: see Minimum-Link Covering Trails for any Hypercubic Lattice.
Lastly, let me just point out how the $4 \times 4 \times \dots \times 4$ case is still unsolved, since I have just shown that the optimal solution is $21$, $22$, or $23$ (see Solving the n_1 × n_2 × n_3 Points Problem for n_3 < 6), and (IMHO) this remains the most interesting open $3$D problem in this family.
