To unlock all Advanced Puzzles, you must first clear the first 60 Basic Puzzles. "Clearing" a puzzle means without using hints; a star marked next to Puzzle Selection states that the puzzle was cleared. However, if you use hints, remember where they are and then restart the puzzle via the Pause Menu. It would make the clearing much easier, and less brain work would be needed.
Successfully complete the 60 starting Basic puzzles without using the in-game hints. Note: When a puzzle is completed with no in-game hints it will be marked with a star icon at the puzzle selection screen. You can still use an in-game hint, then pause game play and restart that puzzle. Finish it the next playthrough without using the in-game hint to mark it completed with the star icon.