Recommend a puzzle game

Here is the google play link: play.google.com/store/apps/details?id=com.lab8.Rect
It's called Rect
Players complete the puzzles by stepping the rects over each other to reach the dashed goal.
From each numbered rect, the player can lay that many rects in one of four directions.
When there are rects already in the path, the next rect "steps over" that rect as it lays out its own.

There are several great puzzle games on the DS. One that springs to mind would be Tetris DS, of course. Tetris is a classic, and the DS version brings even more variety to the table. The touch mode is a lot of fun, and very addicting. Another game you might want to check out is Meteos. This game is kind of like Bejeweled, but at a more intense and addicting pace. This game had be going for a LONG time. There's also, of course, Sudoku games, which are increasingly popular. But for a game similar to Touchmaster? There's a game (It's kind of rare, if you find it, buy it!) called Polarium. It's a fantastic game, but again, it's rare. As for games like Professor Layton, there's two game series' that spring to mind, and that's Phoenix Wright games and Touch Detective. Those point and click adventures (While being even better, in most people's opinions, to Professor Layton) are going to throw you for a loop with their hard to figure out plotlines but they also keep you sucked in until the very end of the games. Those are games that I would most definitely recommend to someone who likes the titles you mentioned.
Source(s):
Work at Gamestop, wikipedia and experience.