This is a more geometric version of the previous question, "Lattice-cube minimal blocking sets". I will first specialize to $\mathbb{R}^3$, $d=3$.
View an $n \times n \times n$ cube $C_3(n)$ as formed of $n^3$ unit cubes glued face-to-face. I would like to find a minimal blocking object $B$ inside $C_3(n)$, which I define as a collection of the unit cubes in $C_3(n)$ with the following two properties: (a) The shadow of $B$ by parallel light rays in the three orthogonal directions (parallel to the cube edges) is an $n \times n$ filled square. (b) $B$ is a connected object, in the sense that its dual graph is connected. Here the dual graph has a node for each unit cube in $B$ and an edge between each pair of cubes that share a face.
$B$ is intended to be a minimal volume object that casts shadows like a cube. The connectedness condition ensures one could build a physical model of $B$.
The previous MO question did not include the connected condition. There it was
shown that blocking sets of size $n^2$ are attainable, $n^{d-1}$ in
dimension $d$.
Certainly that lower bound is no longer achievable in general,
as can be seen with $C_2(2)$:
a $2 \times 2$ square in dimension $d=2$ needs $3$ rather than $2=2^1$
unit squares to form a connected blocking object.
Exploring $n=3$ in $\mathbb{R}^3$, I have been unable to create a connected blocking
object with fewer than 15 cubes:
\[ \left[\begin{array}{ccc} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \end{array}\right] \hspace{0.25 in} \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 0 \end{array}\right] \hspace{0.25in} \left[\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right] \]
Fifteen seems excessive—more than half!. Can anyone see a better solution? In addition, I do not see how to generalize to $C_3(n)$, let alone to $C_d(n)$, the same question in $d$ dimensions.
Addendum. Here is Joel's 13-cube blocker from his answer below:
