Find minimum number of figures needed , so that no additional figure can be added? I have a $6\times 12 $ rectangle, which I need to fill by the following figure:

What is the minimum number of figures I need to use, so that no additional figure can be added? The figure can be rotated but cannot overlap.
How this question can be solved mathematically without trial and error:
Here is one possible answer:


You want to compute the independent domination number of a graph with one node per tromino and an edge for each pair of trominoes that overlap.  You can solve the problem via integer linear programming, as shown in my answer here.  Optimal values for $n \times 2n$ with $n\in\{1,\dots,10\}$ are $0, 2, 3, 6, 9, \color{red}{12}, 17, 22, 28, 35$.
