what game engine do you use for your android game?

Seems like a minority here, but I went for Phaser.io in combination with Crosswalk. But you come pretty close to your limits if you want to go 3D.
Big benefit is of course that it easily runs on almost any platform and you only need Javascript and HTML knowledge ;)
 
Is there a free game engine for simple games like memory games?

There are quite a few. LibGDX, Unity (yes, you can have it for free) or AndEngine. As already posted I use the latter (it is very easy to use), but obviously a lot of people like unity. I do not like it because of the package size (even an empty package is huge).
 
we have used the framework that the book Beginning Android 4 Games Development is based on. However, if we started now we would have used libgdx