The "long comment" by Pietro Majer here suggests the following problem. Let $S$ be a finite subset of $\mathbb{Z}\times \mathbb{Z}$. By a rectangle, I mean an $a\times b$ array of continguous elements of $\mathbb{Z}\times \mathbb{Z}$ ($a,b\geq 1$). E.g., $\{(0,0),(2,0)\}$ is not a rectangle since $(1,0)$ is missing.

(1) Let $\nu(S)$ be the most number of elements of $S$ such that no two lie in a rectangle contained in $S$. Let $\rho(S)$ be the least number of disjoint rectangles whose union is $S$. For what $S$ do we have $\nu(S)=\rho(S)$? Clearly $\nu(S)\leq \rho(S)$.

(2) Suppose we remove a maximal rectangle $R$ from $S$, then a maximal rectangle $R'$ from $S-R$, etc., until after $k$ steps we have removed all the elements of $S$. For what $S$ do we have that $k=\rho(S)$ (for any choice of $R,R',\dots$)?

These questions can easily be extended to higher dimensions.

The "long comment" by Pietro Majer on Reference for puzzle on dividing piles and scoring products suggests the following problem. Let $S$ be a finite subset of $\mathbb{Z}\times \mathbb{Z}$. By a rectangle, I mean an $a\times b$ array of contiguous elements of $\mathbb{Z}\times \mathbb{Z}$ ($a,b\geq 1$). E.g., $\{(0,0),(2,0)\}$ is not a rectangle since $(1,0)$ is missing.

1. Let $\nu(S)$ be the greatest number of elements of $S$ such that no two lie in a rectangle contained in $S$. Let $\rho(S)$ be the least number of disjoint rectangles whose union is $S$. For what $S$ do we have $\nu(S)=\rho(S)$? Clearly $\nu(S)\leq \rho(S)$.

2. Suppose we remove a maximal rectangle $R$ from $S$, then a maximal rectangle $R'$ from $S-R$, etc., until after $k$ steps we have removed all the elements of $S$. For what $S$ do we have that $k=\rho(S)$ (for any choice of $R,R',\dotsc$)?

These questions can easily be extended to higher dimensions.