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You'll have to use some experimentation here, to get a feel for what does and doesn't work.

Do you think you could fill the whole grid using just one shape, if you chose the right 8 cards? What sort of quadrilateral might have lots of nice properties?

It's pretty obvious that you can't come up with a shape that has just 1 axis of symmetry and more than 1 axis of symmetry, so if you put one of these at the start of a row and the other at a start of a column, you know that you won't be able to fill in the square where they intersect. Can you think of any other combinations like this? Is it possible to choose them so that you can't fill in any of the squares?