Tatami

Here is an excerpt from Tatami's App Store description:

Tatami is a casual puzzle game where you fill the board with boxes of the specified size. If you like thinky games like Shikaku, Picross, or Sudoku, but wish for shorter pick-up-and-play levels, you’ll love Tatami.

Things start simple, but as soon as you get comfortable the game gets a little trickier and a little weirder. And then some more! But it stays a kind game: there are no timers or scores, and you can skip any level with no penalty.

And the levels never end! There are enough high quality, procedurally generated boards to last you a lifetime. On the road to infinity, there are over 100 special handcrafted levels full of surprises. And maybe there’s a super secret, super tricky challenge hidden somewhere; I’m not at liberty to say.

The first two hundred levels are free, and the rest can be unlocked with a single in-app purchase. The same purchase allows you to customize the game with themes and cute alternative icons.