Professor Layton and the Diabolical Box plays just like the first game. You control Luke and the Professor by tapping the foot icon on the touch screen (the controls are all touch screen) and examine each environment you come to by tapping on items and people. This is standard DS adventure game controls.
By tapping on things you can get either hint coins, which help you solve puzzles, clues to help you solve the mysteries or a puzzle to solve. If it's a puzzle, a little red exclamation mark pops up.
While the puzzles aren't the same as the first game, they are in the same vein. They will usually have something to do with the dialogue or the environment (albeit loosely) and consist of a physical puzzle, such as moving blocks, a math puzzle (word problems, anyone?) or a brain teaser.
Professor Layton and the Diabolical Box does not go easy on you with the puzzles either. Sure the first ones are simple, but they are luring you into a false sense of security. Before you know it - BAM! - where did that come from? Oh, its just another puzzle, but this time it's a lot worse.
Each puzzle you solve gets you picarats. Every puzzle is worth a certain amount of picarats, and if you solve it with the correct answer the first time you'll get them all. If not, the number will decrease each time you try to solve it and fail.