A puzzle game that looks like a Japanese Pop-Up book, yes please. Tengami is full of texture and requires patience. Your look is that of a woodcut protagonist and you begin by lamenting the loss of a dream.
Tap and travel. It takes a little time for the character to get where he's going, so enjoy the music and scenery. Sometimes pieces of the landscape flip. Sometimes you can ring a chime or open a door. Sometimes there are wolves. Look for glowing auras that guide you towards bits of scenery you can interact with. I am currently stuck at the first challenge of the Mountain Level, where you must create a path by flipping through a series of options. Luckily, there's a walkthrough here, which I may need to consult.
Play it on an iPad. This game is too pretty for a tiny phone screen (I know this because I played it on my phone and felt disappointed that I couldn't see as much of the details as I wanted).
Nyamyam, the company that created Tengami has gotten some recognition and awards for the game, and also has links to more articles and reviews.
This game is too quiet and slow for many an action gamer, but with those who appreciate art and like something a bit more meditative, this game will find a home. Recommend it to your anime club, fans of fairy tales, fans of Yoshitaka Amano (see: Sandman: The Dream Hunters or The Sky: the art of Final Fantasy)' and anyone who has seen the movie Bunraku.