Contemplate the issue from the angle of the coin placer, who needs to put cash in such a method that, irrespective of which $3$-subset of rows and $3$-subset of columns are chosen, a minimum of one coin seems within the complement (a $3 instances 3$ subgrid) and therefore will not be taken. Allow us to discover the minimal variety of cash wanted to fulfill this requirement.
You may clear up the issue by way of integer linear programming as follows. Let binary determination variable $x_{ij}$ point out whether or not cell $(i,j)$ accommodates a coin. The issue is to attenuate the quantity $$sum_{i=1}^6 sum_{j=1}^6 x_{ij}$$ of positioned cash topic to linear (set overlaying) constraints
$$sum_{(i,j)in S} x_{ij} ge 1$$
for all $3 instances 3$ subgrids $S$.
The minimal seems to be
10,
attained, for instance, by
start{matrix} 1 &0 &0 &0 &0 &0 0 &1 &0 &0 &0 &0 0 &0 &1 &0 &1 &0 0 &0 &0 &1 &0 &1 0 &0 &1 &0 &0 &1 0 &0 &0 &1 &1 &0 finish{matrix}
which means that each placement of at most
9
cash can all be taken.
Extra typically, for a $2n instances 2n$ board the biggest variety of cash that may be assured to be taken by way of $n$ rows and $n$ columns is
$3n$
for $nin{1,2,3,4,5}$.
