So, as most Minecraft players quickly learn, the best way to flood an area is to place water along a diagonal. There are of course other ways to flood the same area with the same number of buckets placed.
For those not familiar with the game, we can think of this as a different sort of game. We start with a $0, 1$ matrix of size $n$. At each step, every $0$ with at least two $1$'s directly adjacent to it is replaced with a $1$. Its not hard to see that this process always terminates. The objective is to end with a matrix of all $1$'s with a minimal number of $1$'s to start.
Its not hard to see that an optimal initial arrangement is the identity matrix, and that the minimum number of starting $1$’s is $n$. Of course, there are other matrices that work, such as $$\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix}.$$
We quickly see that we are looking for permutation matrices, but not every permutation matrix works. For example $$\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\end{pmatrix}$$ does not work.
The question then is this: which permutation matrices do work? Is there a nice characterization? We can think of forming "blocks" and then growing blocks together if they share a corner, but is there something else we can say? It appears the counter example given is some kind of forbidden configuration.
Edit: it appears that there are non-permutation matrices that work, such as $$\begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}.$$ Regardless, we can add the assumption that we are working only with permutation matrices. Thus we only consider expansion across corners as the other type of growth cannot happen.
Now if we consider starting with a permuation matrix, and grow it to its final state, we see that the final states are a bunch of disjoint squares of filled in area. We can shrink these squares to a single cell, so equivalently, we ask ourselves which permutation matrices exhibit no growth at all?