Least number of marked symbols

You can solve this set covering problem via integer linear programming as follows. For each pentomino $p$, let $C_p$ be the set of (five) grid cells that comprise it. For each grid cell $(i,j)$, let binary decision variable $x_{i,j}$ indicate whether that cell is marked. The problem is to minimize $sum_{i,j} x_{i,j}$ subject to linear constraints: $$sum_{(i,j)in C_p} x_{i,j} ge 1 quad text{for all $p$}$$ The optimal values for $nin{1,dots,10}$ are begin{matrix} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 hline min & 0 & 0 & 3 & 5 & 8 & 13 & 17 & 24 & 31 & 39 end{matrix}