There's also a free version, which should be identical apart from containing ads.
All puzzle modes are based on tiles whose shape is sort of a cloverleaf; each of the four leaves can be missing. The first puzzle mode uses a 2x2 grid so it acts as a tutorial: it's very easy to solve regardless of the random tiles you are served.
Should you be stuck, there's also a hint option which will place one tile on the board, but can be used only once per puzzle. Also, for some unclear reason, in some puzzles you start with a tile already on the board (which cannot be moved around).
In most of the puzzle modes, tiles also have different colors. For example, you may need to ensure that in every row and column, every color is represented exactly once.
Placing tiles on the board works well enough, but removing them (when you realise you made a mistake) felt a bit cumbersome. There isn't a quick way to pick up a tile and just throw it away with a swipe of the finger; you need to move it all the way down to one of the empty holes below the board. If you move it out of the board but not over an empty hole, it snaps back on the board.
The App Store description says that Perfect Tiling is "a game of logic and observation". This is a funny conincidence since it's exactly the same incipit that I used for Twin Beams. :-)
To be honest, in some of the puzzle modes logic doesn't seem to play a big enough part; it looks like all you can possibly do is place the first tiles almost at random, then attempt to fill in the gaps.
There are, however, at least a couple of puzzle modes where logic deductions can be proficiently applied; those are the modes where the colors of the tiles needs to match colors already placed on the board, significantly reducing the possible arrangements. Like in the fourth mode:
The fifth puzzle mode seems to be the one where this works best:
The routine protanopia simulation shows that some improvements could be made using a slightly different palette and/or shapes other than circles to identify the tiles:
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