  A $d$-dimensional brick is a set $I_1\times \cdots \times I_d$ where each
$I_i$ is an interval. Given a brick $B$, a brick partition of $B$ is a
partition of $B$ into bricks. A brick partition $\mathcal{P}_d$ of a
$d$-dimensional brick is $k$-piercing if every axis-parallel line intersects at
least $k$ bricks in $\mathcal{P}_d$. Bucic et al. explicitly asked the minimum
size $p(d, k)$ of a $k$-piercing brick partition of a $d$-dimensional brick.
The answer is known to be $4(k-1)$ when $d=2$. Our first result almost
determines $p(3, k)$. Namely, we construct a $k$-piercing brick partition of a
$3$-dimensional brick with $12k-15$ parts, which is off by only $1$ from the
known lower bound. As a generalization of the above question, we also seek the
minimum size $s(d, k)$ of a brick partition $\mathcal{P}_d$ of a
$d$-dimensional brick where each axis-parallel plane intersects at least $k$
bricks in $\mathcal{P}_d$. We resolve the question in the $3$-dimensional case
by determining $s(3, k)$ for all $k$.
