Mid 2014 I was introduced to the generalization of a set of shapes we are all aware of. For n=2, we call them dominos, for n=4 we call them Tetris pieces.
The shapes you converge on a given number of squares contained are called Polyominos. It is pretty easy to work your way through up to about n=5 by hand, but the number of unique entries goes up quite quickly. Images for n=8 and n=10 are below.
Beyond 10 modern browsers begin to struggle to render correctly. The last image in this set is for n=14. This is a 200 megapixel image, so give extend your computer some sympathy.