Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$. Define a blocking set for a lattice cube to be a set of points in the cube that meets every axis-parallel line through a lattice point of any facet of the cube. Let $b_d(n)$ be the minimum cardinality of a blocking set for $C_d(n)$.

Q1. What is the function $b_d(n)$?

For $d=2$, selecting the $n$ points on the diagonal through $(1,1)$ and $(n,n)$ suffices to block the $n$ lines through each facet/side of the cube/square; so $C_2(n)=n$. Because a facet consists of $n^{d-1}$ points, and a line orthogonal to the facet through each point must be blocked, $n^{d-1}$ is a lower bound on $b_d(n)$. A more specific version of Q1 is:

Q2. Can the lower bound be achieved? Is $b_d(n) = n^{d-1}$? If the lower bound cannot always be achieved, when can it be achieved?

For $C_3(3)$, the $\mathbb{Z}^3$-cube consisting of $27$ points, here is a blocking set $B$ with $|B|=10=n^2+1$:

Perhaps these questions have already been explored? If so, I would appreciate references. Thanks!

My interest in the above questions derive from the more geometrical question: What is the "minimal object'' that casts a (hyper)cube-shadow from parallel light rays in the $d$ coordinate directions? I defer this question until I understand the combinatorial version.

Addendum. As per Johan Wästlund and Eoin, indeed the lowerbound is achievable. Here is a 9-point blocking set for $C_3(3)$:

2 edited body; added 12 characters in body

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each coorindate of its $d$ coorindates in $\lbrace 1,2,\ldots,d 1,2,\ldots,n \rbrace$. Define a blocking set for a lattice cube to be a set of points in the cube that meets every axis-parallel line through a lattice point of any facet of the cube. Let $b_d(n)$ be the minimum cardinality of a blocking set for $C_d(n)$.

Q1. What is the function $b_d(n)$?

For $d=2$, selecting the $n$ points on the diagonal through $(1,1)$ and $(n,n)$ suffices to block the $n$ lines through each facet/side of the cube/square; so $C_2(n)=n$. Because a facet consists of $n^{d-1}$ points, and a line orthogonal to the facet through each point must be blocked, $n^{d-1}$ is a lower bound on $b_d(n)$. A more specific version of Q1 is:

Q2. Can the lower bound be achieved? Is $b_d(n) = n^{d-1}$? If the lower bound cannot always be achieved, when can it be achieved?

For $C_3(3)$, the $\mathbb{Z}^3$-cube consisting of $27$ points, here is a blocking set $B$ with $|B|=10=n^2+1$:

Perhaps these questions have already been explored? If so, I would appreciate references. Thanks!

My interest in the above questions derive from the more geometrical question: What is the "minimal object'' that casts a (hyper)cube-shadow from parallel light rays in the $d$ coordinate directions? I defer this question until I understand the combinatorial version.

1

# Lattice-cube minimal blocking sets

Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with each coorindate in $\lbrace 1,2,\ldots,d \rbrace$. Define a blocking set for a lattice cube to be a set of points in the cube that meets every axis-parallel line through a lattice point of any facet of the cube. Let $b_d(n)$ be the minimum cardinality of a blocking set for $C_d(n)$.

Q1. What is the function $b_d(n)$?

For $d=2$, selecting the $n$ points on the diagonal through $(1,1)$ and $(n,n)$ suffices to block the $n$ lines through each facet/side of the cube/square; so $C_2(n)=n$. Because a facet consists of $n^{d-1}$ points, and a line orthogonal to the facet through each point must be blocked, $n^{d-1}$ is a lower bound on $b_d(n)$. A more specific version of Q1 is:

Q2. Can the lower bound be achieved? Is $b_d(n) = n^{d-1}$? If the lower bound cannot always be achieved, when can it be achieved?

For $C_3(3)$, the $\mathbb{Z}^3$-cube consisting of $27$ points, here is a blocking set $B$ with $|B|=10=n^2+1$:

Perhaps these questions have already been explored? If so, I would appreciate references. Thanks!

My interest in the above questions derive from the more geometrical question: What is the "minimal object'' that casts a (hyper)cube-shadow from parallel light rays in the $d$ coordinate directions? I defer this question until I understand the combinatorial version.