rsl12

Ok, assuming that you have to use all shapes, here's an extra optimization, related to something else we talked about before: if you color the board as a checkerboard, above you have, say, 5 black and 4 white squares. The number of blocks on black squares has to be 4+2n, while the number of blocks on white squares has to be 1+2n. (As a shortcut, it's probably easier just to say the mod 2 of the number of blocks on black squares is 0, and the mod 2 on the white squares is 1). Now we can assign a pair of numbers for each shape that will be used, depending on the number of white/black squares needed to cover it (order is arbitrary). So for the example above, we have:

2,1
2,1
2,2
3,1
3,3
1,0
1,1
2,2

All of these ordered pairs can be flipped, but the X column has to add up such that the mod 2 of the number is 0, while the sum of the Y column has to have a mod 2 of 1. Does that reduce any time?

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